src/HOL/Ring_and_Field.thy
author paulson
Wed Aug 15 12:52:56 2007 +0200 (2007-08-15)
changeset 24286 7619080e49f0
parent 23879 4776af8be741
child 24422 c0b5ff9e9e4d
permissions -rw-r--r--
ATP blacklisting is now in theory data, attribute noatp
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(*  Title:   HOL/Ring_and_Field.thy
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    ID:      $Id$
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    Author:  Gertrud Bauer, Steven Obua, Tobias Nipkow, Lawrence C Paulson, and Markus Wenzel,
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             with contributions by Jeremy Avigad
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*)
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header {* (Ordered) Rings and Fields *}
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theory Ring_and_Field
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imports OrderedGroup
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begin
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text {*
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  The theory of partially ordered rings is taken from the books:
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  \begin{itemize}
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  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
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  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
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  \end{itemize}
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  Most of the used notions can also be looked up in 
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  \begin{itemize}
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  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
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  \item \emph{Algebra I} by van der Waerden, Springer.
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  \end{itemize}
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*}
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class semiring = ab_semigroup_add + semigroup_mult +
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  assumes left_distrib: "(a \<^loc>+ b) \<^loc>* c = a \<^loc>* c \<^loc>+ b \<^loc>* c"
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  assumes right_distrib: "a \<^loc>* (b \<^loc>+ c) = a \<^loc>* b \<^loc>+ a \<^loc>* c"
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class mult_zero = times + zero +
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  assumes mult_zero_left [simp]: "\<^loc>0 \<^loc>* a = \<^loc>0"
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  assumes mult_zero_right [simp]: "a \<^loc>* \<^loc>0 = \<^loc>0"
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class semiring_0 = semiring + comm_monoid_add + mult_zero
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class semiring_0_cancel = semiring + comm_monoid_add + cancel_ab_semigroup_add
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instance semiring_0_cancel \<subseteq> semiring_0
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proof
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  fix a :: 'a
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  have "0 * a + 0 * a = 0 * a + 0"
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    by (simp add: left_distrib [symmetric])
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  thus "0 * a = 0"
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    by (simp only: add_left_cancel)
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  have "a * 0 + a * 0 = a * 0 + 0"
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    by (simp add: right_distrib [symmetric])
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  thus "a * 0 = 0"
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    by (simp only: add_left_cancel)
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qed
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class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
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  assumes distrib: "(a \<^loc>+ b) \<^loc>* c = a \<^loc>* c \<^loc>+ b \<^loc>* c"
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instance comm_semiring \<subseteq> semiring
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proof
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  fix a b c :: 'a
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  show "(a + b) * c = a * c + b * c" by (simp add: distrib)
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  have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
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  also have "... = b * a + c * a" by (simp only: distrib)
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  also have "... = a * b + a * c" by (simp add: mult_ac)
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  finally show "a * (b + c) = a * b + a * c" by blast
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qed
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class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
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instance comm_semiring_0 \<subseteq> semiring_0 ..
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class comm_semiring_0_cancel = comm_semiring + comm_monoid_add + cancel_ab_semigroup_add
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instance comm_semiring_0_cancel \<subseteq> semiring_0_cancel ..
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instance comm_semiring_0_cancel \<subseteq> comm_semiring_0 ..
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class zero_neq_one = zero + one +
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  assumes zero_neq_one [simp]: "\<^loc>0 \<noteq> \<^loc>1"
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class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
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class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult
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  (*previously almost_semiring*)
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instance comm_semiring_1 \<subseteq> semiring_1 ..
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class no_zero_divisors = zero + times +
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  assumes no_zero_divisors: "a \<noteq> \<^loc>0 \<Longrightarrow> b \<noteq> \<^loc>0 \<Longrightarrow> a \<^loc>* b \<noteq> \<^loc>0"
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class semiring_1_cancel = semiring + comm_monoid_add + zero_neq_one
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  + cancel_ab_semigroup_add + monoid_mult
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instance semiring_1_cancel \<subseteq> semiring_0_cancel ..
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instance semiring_1_cancel \<subseteq> semiring_1 ..
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class comm_semiring_1_cancel = comm_semiring + comm_monoid_add + comm_monoid_mult
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  + zero_neq_one + cancel_ab_semigroup_add
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instance comm_semiring_1_cancel \<subseteq> semiring_1_cancel ..
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instance comm_semiring_1_cancel \<subseteq> comm_semiring_0_cancel ..
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instance comm_semiring_1_cancel \<subseteq> comm_semiring_1 ..
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class ring = semiring + ab_group_add
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instance ring \<subseteq> semiring_0_cancel ..
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class comm_ring = comm_semiring + ab_group_add
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instance comm_ring \<subseteq> ring ..
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instance comm_ring \<subseteq> comm_semiring_0_cancel ..
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class ring_1 = ring + zero_neq_one + monoid_mult
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instance ring_1 \<subseteq> semiring_1_cancel ..
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class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
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  (*previously ring*)
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instance comm_ring_1 \<subseteq> ring_1 ..
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instance comm_ring_1 \<subseteq> comm_semiring_1_cancel ..
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class ring_no_zero_divisors = ring + no_zero_divisors
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class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
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class idom = comm_ring_1 + no_zero_divisors
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instance idom \<subseteq> ring_1_no_zero_divisors ..
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class division_ring = ring_1 + inverse +
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  assumes left_inverse [simp]:  "a \<noteq> \<^loc>0 \<Longrightarrow> inverse a \<^loc>* a = \<^loc>1"
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  assumes right_inverse [simp]: "a \<noteq> \<^loc>0 \<Longrightarrow> a \<^loc>* inverse a = \<^loc>1"
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instance division_ring \<subseteq> ring_1_no_zero_divisors
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proof
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  fix a b :: 'a
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  assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
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  show "a * b \<noteq> 0"
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  proof
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    assume ab: "a * b = 0"
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    hence "0 = inverse a * (a * b) * inverse b"
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      by simp
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    also have "\<dots> = (inverse a * a) * (b * inverse b)"
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      by (simp only: mult_assoc)
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    also have "\<dots> = 1"
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      using a b by simp
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    finally show False
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      by simp
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  qed
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qed
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class field = comm_ring_1 + inverse +
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  assumes field_inverse:  "a \<noteq> 0 \<Longrightarrow> inverse a \<^loc>* a = \<^loc>1"
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  assumes divide_inverse: "a \<^loc>/ b = a \<^loc>* inverse b"
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instance field \<subseteq> division_ring
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proof
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  fix a :: 'a
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  assume "a \<noteq> 0"
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  thus "inverse a * a = 1" by (rule field_inverse)
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  thus "a * inverse a = 1" by (simp only: mult_commute)
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qed
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instance field \<subseteq> idom ..
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class division_by_zero = zero + inverse +
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  assumes inverse_zero [simp]: "inverse \<^loc>0 = \<^loc>0"
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subsection {* Distribution rules *}
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text{*For the @{text combine_numerals} simproc*}
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lemma combine_common_factor:
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     "a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)"
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by (simp add: left_distrib add_ac)
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lemma minus_mult_left: "- (a * b) = (-a) * (b::'a::ring)"
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apply (rule equals_zero_I)
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apply (simp add: left_distrib [symmetric]) 
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done
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lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)"
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apply (rule equals_zero_I)
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apply (simp add: right_distrib [symmetric]) 
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done
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lemma minus_mult_minus [simp]: "(- a) * (- b) = a * (b::'a::ring)"
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  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
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lemma minus_mult_commute: "(- a) * b = a * (- b::'a::ring)"
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  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
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lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)"
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by (simp add: right_distrib diff_minus 
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              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma left_diff_distrib: "(a - b) * c = a * c - b * (c::'a::ring)"
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by (simp add: left_distrib diff_minus 
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              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemmas ring_distribs =
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  right_distrib left_distrib left_diff_distrib right_diff_distrib
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text{*This list of rewrites simplifies ring terms by multiplying
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everything out and bringing sums and products into a canonical form
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(by ordered rewriting). As a result it decides ring equalities but
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also helps with inequalities. *}
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lemmas ring_simps = group_simps ring_distribs
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class mult_mono = times + zero + ord +
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  assumes mult_left_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> c \<^loc>* a \<sqsubseteq> c \<^loc>* b"
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  assumes mult_right_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> a \<^loc>* c \<sqsubseteq> b \<^loc>* c"
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class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add 
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class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add
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  + semiring + comm_monoid_add + cancel_ab_semigroup_add
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instance pordered_cancel_semiring \<subseteq> semiring_0_cancel ..
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instance pordered_cancel_semiring \<subseteq> pordered_semiring .. 
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class ordered_semiring = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + mult_mono
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instance ordered_semiring \<subseteq> pordered_cancel_semiring ..
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class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add +
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  assumes mult_strict_left_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> c \<^loc>* a \<sqsubset> c \<^loc>* b"
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  assumes mult_strict_right_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> a \<^loc>* c \<sqsubset> b \<^loc>* c"
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instance ordered_semiring_strict \<subseteq> semiring_0_cancel ..
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instance ordered_semiring_strict \<subseteq> ordered_semiring
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proof
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  fix a b c :: 'a
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  assume A: "a \<le> b" "0 \<le> c"
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  from A show "c * a \<le> c * b"
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    unfolding order_le_less
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    using mult_strict_left_mono by auto
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  from A show "a * c \<le> b * c"
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    unfolding order_le_less
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    using mult_strict_right_mono by auto
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qed
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class mult_mono1 = times + zero + ord +
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  assumes mult_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> c \<^loc>* a \<sqsubseteq> c \<^loc>* b"
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class pordered_comm_semiring = comm_semiring_0
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  + pordered_ab_semigroup_add + mult_mono1
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class pordered_cancel_comm_semiring = comm_semiring_0_cancel
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  + pordered_ab_semigroup_add + mult_mono1
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instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring ..
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class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add +
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  assumes mult_strict_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> c \<^loc>* a \<sqsubset> c \<^loc>* b"
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instance pordered_comm_semiring \<subseteq> pordered_semiring
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proof
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  fix a b c :: 'a
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  assume "a \<le> b" "0 \<le> c"
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  thus "c * a \<le> c * b" by (rule mult_mono)
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  thus "a * c \<le> b * c" by (simp only: mult_commute)
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qed
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instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring ..
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instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict
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proof
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  fix a b c :: 'a
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  assume "a < b" "0 < c"
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  thus "c * a < c * b" by (rule mult_strict_mono)
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  thus "a * c < b * c" by (simp only: mult_commute)
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qed
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instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring
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proof
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  fix a b c :: 'a
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  assume "a \<le> b" "0 \<le> c"
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  thus "c * a \<le> c * b"
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    unfolding order_le_less
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    using mult_strict_mono by auto
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qed
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class pordered_ring = ring + pordered_cancel_semiring 
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instance pordered_ring \<subseteq> pordered_ab_group_add ..
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class lordered_ring = pordered_ring + lordered_ab_group_abs
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instance lordered_ring \<subseteq> lordered_ab_group_meet ..
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instance lordered_ring \<subseteq> lordered_ab_group_join ..
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class abs_if = minus + ord + zero + abs +
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  assumes abs_if: "abs a = (if a \<sqsubset> 0 then (uminus a) else a)"
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(* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors.
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   Basically, ordered_ring + no_zero_divisors = ordered_ring_strict.
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 *)
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class ordered_ring = ring + ordered_semiring + lordered_ab_group + abs_if
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instance ordered_ring \<subseteq> lordered_ring
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proof
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  fix x :: 'a
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  show "\<bar>x\<bar> = sup x (- x)"
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    by (simp only: abs_if sup_eq_if)
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qed
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class ordered_ring_strict = ring + ordered_semiring_strict + lordered_ab_group + abs_if
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instance ordered_ring_strict \<subseteq> ordered_ring ..
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class pordered_comm_ring = comm_ring + pordered_comm_semiring
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instance pordered_comm_ring \<subseteq> pordered_ring ..
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instance pordered_comm_ring \<subseteq> pordered_cancel_comm_semiring ..
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class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict +
haftmann@22390
   325
  (*previously ordered_semiring*)
haftmann@22390
   326
  assumes zero_less_one [simp]: "\<^loc>0 \<sqsubset> \<^loc>1"
paulson@14270
   327
obua@23521
   328
class ordered_idom = comm_ring_1 + ordered_comm_semiring_strict + lordered_ab_group + abs_if
haftmann@22390
   329
  (*previously ordered_ring*)
paulson@14270
   330
obua@14738
   331
instance ordered_idom \<subseteq> ordered_ring_strict ..
paulson@14272
   332
huffman@23073
   333
instance ordered_idom \<subseteq> pordered_comm_ring ..
huffman@23073
   334
haftmann@22390
   335
class ordered_field = field + ordered_idom
paulson@14272
   336
nipkow@15923
   337
lemmas linorder_neqE_ordered_idom =
nipkow@15923
   338
 linorder_neqE[where 'a = "?'b::ordered_idom"]
nipkow@15923
   339
paulson@14272
   340
lemma eq_add_iff1:
nipkow@23477
   341
  "(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"
nipkow@23477
   342
by (simp add: ring_simps)
paulson@14272
   343
paulson@14272
   344
lemma eq_add_iff2:
nipkow@23477
   345
  "(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"
nipkow@23477
   346
by (simp add: ring_simps)
paulson@14272
   347
paulson@14272
   348
lemma less_add_iff1:
nipkow@23477
   349
  "(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))"
nipkow@23477
   350
by (simp add: ring_simps)
paulson@14272
   351
paulson@14272
   352
lemma less_add_iff2:
nipkow@23477
   353
  "(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))"
nipkow@23477
   354
by (simp add: ring_simps)
paulson@14272
   355
paulson@14272
   356
lemma le_add_iff1:
nipkow@23477
   357
  "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))"
nipkow@23477
   358
by (simp add: ring_simps)
paulson@14272
   359
paulson@14272
   360
lemma le_add_iff2:
nipkow@23477
   361
  "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))"
nipkow@23477
   362
by (simp add: ring_simps)
paulson@14272
   363
wenzelm@23389
   364
paulson@14270
   365
subsection {* Ordering Rules for Multiplication *}
paulson@14270
   366
paulson@14348
   367
lemma mult_left_le_imp_le:
nipkow@23477
   368
  "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
nipkow@23477
   369
by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])
paulson@14348
   370
 
paulson@14348
   371
lemma mult_right_le_imp_le:
nipkow@23477
   372
  "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
nipkow@23477
   373
by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])
paulson@14348
   374
paulson@14348
   375
lemma mult_left_less_imp_less:
obua@23521
   376
  "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
nipkow@23477
   377
by (force simp add: mult_left_mono linorder_not_le [symmetric])
paulson@14348
   378
 
paulson@14348
   379
lemma mult_right_less_imp_less:
obua@23521
   380
  "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
nipkow@23477
   381
by (force simp add: mult_right_mono linorder_not_le [symmetric])
paulson@14348
   382
paulson@14265
   383
lemma mult_strict_left_mono_neg:
nipkow@23477
   384
  "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)"
paulson@14265
   385
apply (drule mult_strict_left_mono [of _ _ "-c"])
paulson@14265
   386
apply (simp_all add: minus_mult_left [symmetric]) 
paulson@14265
   387
done
paulson@14265
   388
obua@14738
   389
lemma mult_left_mono_neg:
nipkow@23477
   390
  "[|b \<le> a; c \<le> 0|] ==> c * a \<le>  c * (b::'a::pordered_ring)"
obua@14738
   391
apply (drule mult_left_mono [of _ _ "-c"])
obua@14738
   392
apply (simp_all add: minus_mult_left [symmetric]) 
obua@14738
   393
done
obua@14738
   394
paulson@14265
   395
lemma mult_strict_right_mono_neg:
nipkow@23477
   396
  "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)"
paulson@14265
   397
apply (drule mult_strict_right_mono [of _ _ "-c"])
paulson@14265
   398
apply (simp_all add: minus_mult_right [symmetric]) 
paulson@14265
   399
done
paulson@14265
   400
obua@14738
   401
lemma mult_right_mono_neg:
nipkow@23477
   402
  "[|b \<le> a; c \<le> 0|] ==> a * c \<le>  (b::'a::pordered_ring) * c"
obua@14738
   403
apply (drule mult_right_mono [of _ _ "-c"])
obua@14738
   404
apply (simp)
obua@14738
   405
apply (simp_all add: minus_mult_right [symmetric]) 
obua@14738
   406
done
paulson@14265
   407
wenzelm@23389
   408
paulson@14265
   409
subsection{* Products of Signs *}
paulson@14265
   410
avigad@16775
   411
lemma mult_pos_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"
paulson@14265
   412
by (drule mult_strict_left_mono [of 0 b], auto)
paulson@14265
   413
avigad@16775
   414
lemma mult_nonneg_nonneg: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b"
obua@14738
   415
by (drule mult_left_mono [of 0 b], auto)
obua@14738
   416
obua@14738
   417
lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0"
paulson@14265
   418
by (drule mult_strict_left_mono [of b 0], auto)
paulson@14265
   419
avigad@16775
   420
lemma mult_nonneg_nonpos: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0"
obua@14738
   421
by (drule mult_left_mono [of b 0], auto)
obua@14738
   422
obua@14738
   423
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" 
obua@14738
   424
by (drule mult_strict_right_mono[of b 0], auto)
obua@14738
   425
avigad@16775
   426
lemma mult_nonneg_nonpos2: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" 
obua@14738
   427
by (drule mult_right_mono[of b 0], auto)
obua@14738
   428
avigad@16775
   429
lemma mult_neg_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"
paulson@14265
   430
by (drule mult_strict_right_mono_neg, auto)
paulson@14265
   431
avigad@16775
   432
lemma mult_nonpos_nonpos: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b"
obua@14738
   433
by (drule mult_right_mono_neg[of a 0 b ], auto)
obua@14738
   434
paulson@14341
   435
lemma zero_less_mult_pos:
obua@14738
   436
     "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
haftmann@21328
   437
apply (cases "b\<le>0") 
paulson@14265
   438
 apply (auto simp add: order_le_less linorder_not_less)
paulson@14265
   439
apply (drule_tac mult_pos_neg [of a b]) 
paulson@14265
   440
 apply (auto dest: order_less_not_sym)
paulson@14265
   441
done
paulson@14265
   442
obua@14738
   443
lemma zero_less_mult_pos2:
obua@14738
   444
     "[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
haftmann@21328
   445
apply (cases "b\<le>0") 
obua@14738
   446
 apply (auto simp add: order_le_less linorder_not_less)
obua@14738
   447
apply (drule_tac mult_pos_neg2 [of a b]) 
obua@14738
   448
 apply (auto dest: order_less_not_sym)
obua@14738
   449
done
obua@14738
   450
paulson@14265
   451
lemma zero_less_mult_iff:
obua@14738
   452
     "((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
   453
apply (auto simp add: order_le_less linorder_not_less mult_pos_pos 
avigad@16775
   454
  mult_neg_neg)
paulson@14265
   455
apply (blast dest: zero_less_mult_pos) 
obua@14738
   456
apply (blast dest: zero_less_mult_pos2)
paulson@14265
   457
done
paulson@14265
   458
huffman@22990
   459
lemma mult_eq_0_iff [simp]:
huffman@22990
   460
  fixes a b :: "'a::ring_no_zero_divisors"
huffman@22990
   461
  shows "(a * b = 0) = (a = 0 \<or> b = 0)"
huffman@22990
   462
by (cases "a = 0 \<or> b = 0", auto dest: no_zero_divisors)
huffman@22990
   463
huffman@22990
   464
instance ordered_ring_strict \<subseteq> ring_no_zero_divisors
huffman@22990
   465
apply intro_classes
paulson@14265
   466
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
paulson@14265
   467
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
paulson@14265
   468
done
paulson@14265
   469
paulson@14265
   470
lemma zero_le_mult_iff:
obua@14738
   471
     "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
paulson@14265
   472
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
paulson@14265
   473
                   zero_less_mult_iff)
paulson@14265
   474
paulson@14265
   475
lemma mult_less_0_iff:
obua@14738
   476
     "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"
paulson@14265
   477
apply (insert zero_less_mult_iff [of "-a" b]) 
paulson@14265
   478
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   479
done
paulson@14265
   480
paulson@14265
   481
lemma mult_le_0_iff:
obua@14738
   482
     "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
paulson@14265
   483
apply (insert zero_le_mult_iff [of "-a" b]) 
paulson@14265
   484
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   485
done
paulson@14265
   486
obua@14738
   487
lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b) | (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)"
avigad@16775
   488
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
obua@14738
   489
obua@14738
   490
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> (0::_::pordered_cancel_semiring)" 
avigad@16775
   491
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
obua@14738
   492
obua@23095
   493
lemma zero_le_square[simp]: "(0::'a::ordered_ring_strict) \<le> a*a"
obua@23095
   494
by (simp add: zero_le_mult_iff linorder_linear)
obua@23095
   495
obua@23095
   496
lemma not_square_less_zero[simp]: "\<not> (a * a < (0::'a::ordered_ring_strict))"
obua@23095
   497
by (simp add: not_less)
paulson@14265
   498
obua@14738
   499
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
obua@14738
   500
      theorems available to members of @{term ordered_idom} *}
obua@14738
   501
obua@14738
   502
instance ordered_idom \<subseteq> ordered_semidom
paulson@14421
   503
proof
paulson@14421
   504
  have "(0::'a) \<le> 1*1" by (rule zero_le_square)
paulson@14430
   505
  thus "(0::'a) < 1" by (simp add: order_le_less) 
paulson@14421
   506
qed
paulson@14421
   507
obua@14738
   508
instance ordered_idom \<subseteq> idom ..
obua@14738
   509
paulson@14387
   510
text{*All three types of comparision involving 0 and 1 are covered.*}
paulson@14387
   511
paulson@17085
   512
lemmas one_neq_zero = zero_neq_one [THEN not_sym]
paulson@17085
   513
declare one_neq_zero [simp]
paulson@14387
   514
obua@14738
   515
lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"
paulson@14268
   516
  by (rule zero_less_one [THEN order_less_imp_le]) 
paulson@14268
   517
obua@14738
   518
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"
obua@14738
   519
by (simp add: linorder_not_le) 
paulson@14387
   520
obua@14738
   521
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"
obua@14738
   522
by (simp add: linorder_not_less) 
paulson@14268
   523
wenzelm@23389
   524
paulson@14268
   525
subsection{*More Monotonicity*}
paulson@14268
   526
paulson@14268
   527
text{*Strict monotonicity in both arguments*}
paulson@14268
   528
lemma mult_strict_mono:
obua@14738
   529
     "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
haftmann@21328
   530
apply (cases "c=0")
avigad@16775
   531
 apply (simp add: mult_pos_pos) 
paulson@14268
   532
apply (erule mult_strict_right_mono [THEN order_less_trans])
paulson@14268
   533
 apply (force simp add: order_le_less) 
paulson@14268
   534
apply (erule mult_strict_left_mono, assumption)
paulson@14268
   535
done
paulson@14268
   536
paulson@14268
   537
text{*This weaker variant has more natural premises*}
paulson@14268
   538
lemma mult_strict_mono':
obua@14738
   539
     "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
paulson@14268
   540
apply (rule mult_strict_mono)
paulson@14268
   541
apply (blast intro: order_le_less_trans)+
paulson@14268
   542
done
paulson@14268
   543
paulson@14268
   544
lemma mult_mono:
paulson@14268
   545
     "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 
obua@14738
   546
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
paulson@14268
   547
apply (erule mult_right_mono [THEN order_trans], assumption)
paulson@14268
   548
apply (erule mult_left_mono, assumption)
paulson@14268
   549
done
paulson@14268
   550
huffman@21258
   551
lemma mult_mono':
huffman@21258
   552
     "[|a \<le> b; c \<le> d; 0 \<le> a; 0 \<le> c|] 
huffman@21258
   553
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
huffman@21258
   554
apply (rule mult_mono)
huffman@21258
   555
apply (fast intro: order_trans)+
huffman@21258
   556
done
huffman@21258
   557
obua@14738
   558
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"
paulson@14387
   559
apply (insert mult_strict_mono [of 1 m 1 n]) 
paulson@14430
   560
apply (simp add:  order_less_trans [OF zero_less_one]) 
paulson@14387
   561
done
paulson@14387
   562
avigad@16775
   563
lemma mult_less_le_imp_less: "(a::'a::ordered_semiring_strict) < b ==>
avigad@16775
   564
    c <= d ==> 0 <= a ==> 0 < c ==> a * c < b * d"
avigad@16775
   565
  apply (subgoal_tac "a * c < b * c")
avigad@16775
   566
  apply (erule order_less_le_trans)
avigad@16775
   567
  apply (erule mult_left_mono)
avigad@16775
   568
  apply simp
avigad@16775
   569
  apply (erule mult_strict_right_mono)
avigad@16775
   570
  apply assumption
avigad@16775
   571
done
avigad@16775
   572
avigad@16775
   573
lemma mult_le_less_imp_less: "(a::'a::ordered_semiring_strict) <= b ==>
avigad@16775
   574
    c < d ==> 0 < a ==> 0 <= c ==> a * c < b * d"
avigad@16775
   575
  apply (subgoal_tac "a * c <= b * c")
avigad@16775
   576
  apply (erule order_le_less_trans)
avigad@16775
   577
  apply (erule mult_strict_left_mono)
avigad@16775
   578
  apply simp
avigad@16775
   579
  apply (erule mult_right_mono)
avigad@16775
   580
  apply simp
avigad@16775
   581
done
avigad@16775
   582
wenzelm@23389
   583
paulson@14268
   584
subsection{*Cancellation Laws for Relationships With a Common Factor*}
paulson@14268
   585
paulson@14268
   586
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
paulson@14268
   587
   also with the relations @{text "\<le>"} and equality.*}
paulson@14268
   588
paulson@15234
   589
text{*These ``disjunction'' versions produce two cases when the comparison is
paulson@15234
   590
 an assumption, but effectively four when the comparison is a goal.*}
paulson@15234
   591
paulson@15234
   592
lemma mult_less_cancel_right_disj:
obua@14738
   593
    "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
haftmann@21328
   594
apply (cases "c = 0")
paulson@14268
   595
apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
paulson@14268
   596
                      mult_strict_right_mono_neg)
paulson@14268
   597
apply (auto simp add: linorder_not_less 
paulson@14268
   598
                      linorder_not_le [symmetric, of "a*c"]
paulson@14268
   599
                      linorder_not_le [symmetric, of a])
paulson@14268
   600
apply (erule_tac [!] notE)
paulson@14268
   601
apply (auto simp add: order_less_imp_le mult_right_mono 
paulson@14268
   602
                      mult_right_mono_neg)
paulson@14268
   603
done
paulson@14268
   604
paulson@15234
   605
lemma mult_less_cancel_left_disj:
obua@14738
   606
    "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
haftmann@21328
   607
apply (cases "c = 0")
obua@14738
   608
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
obua@14738
   609
                      mult_strict_left_mono_neg)
obua@14738
   610
apply (auto simp add: linorder_not_less 
obua@14738
   611
                      linorder_not_le [symmetric, of "c*a"]
obua@14738
   612
                      linorder_not_le [symmetric, of a])
obua@14738
   613
apply (erule_tac [!] notE)
obua@14738
   614
apply (auto simp add: order_less_imp_le mult_left_mono 
obua@14738
   615
                      mult_left_mono_neg)
obua@14738
   616
done
paulson@14268
   617
paulson@15234
   618
paulson@15234
   619
text{*The ``conjunction of implication'' lemmas produce two cases when the
paulson@15234
   620
comparison is a goal, but give four when the comparison is an assumption.*}
paulson@15234
   621
paulson@15234
   622
lemma mult_less_cancel_right:
paulson@15234
   623
  fixes c :: "'a :: ordered_ring_strict"
paulson@15234
   624
  shows      "(a*c < b*c) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
paulson@15234
   625
by (insert mult_less_cancel_right_disj [of a c b], auto)
paulson@15234
   626
paulson@15234
   627
lemma mult_less_cancel_left:
paulson@15234
   628
  fixes c :: "'a :: ordered_ring_strict"
paulson@15234
   629
  shows      "(c*a < c*b) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
paulson@15234
   630
by (insert mult_less_cancel_left_disj [of c a b], auto)
paulson@15234
   631
paulson@14268
   632
lemma mult_le_cancel_right:
obua@14738
   633
     "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
paulson@15234
   634
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj)
paulson@14268
   635
paulson@14268
   636
lemma mult_le_cancel_left:
obua@14738
   637
     "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
paulson@15234
   638
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj)
paulson@14268
   639
paulson@14268
   640
lemma mult_less_imp_less_left:
paulson@14341
   641
      assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
obua@14738
   642
      shows "a < (b::'a::ordered_semiring_strict)"
paulson@14377
   643
proof (rule ccontr)
paulson@14377
   644
  assume "~ a < b"
paulson@14377
   645
  hence "b \<le> a" by (simp add: linorder_not_less)
wenzelm@23389
   646
  hence "c*b \<le> c*a" using nonneg by (rule mult_left_mono)
paulson@14377
   647
  with this and less show False 
paulson@14377
   648
    by (simp add: linorder_not_less [symmetric])
paulson@14377
   649
qed
paulson@14268
   650
paulson@14268
   651
lemma mult_less_imp_less_right:
obua@14738
   652
  assumes less: "a*c < b*c" and nonneg: "0 <= c"
obua@14738
   653
  shows "a < (b::'a::ordered_semiring_strict)"
obua@14738
   654
proof (rule ccontr)
obua@14738
   655
  assume "~ a < b"
obua@14738
   656
  hence "b \<le> a" by (simp add: linorder_not_less)
wenzelm@23389
   657
  hence "b*c \<le> a*c" using nonneg by (rule mult_right_mono)
obua@14738
   658
  with this and less show False 
obua@14738
   659
    by (simp add: linorder_not_less [symmetric])
obua@14738
   660
qed  
paulson@14268
   661
paulson@14268
   662
text{*Cancellation of equalities with a common factor*}
paulson@24286
   663
lemma mult_cancel_right [simp,noatp]:
huffman@22990
   664
  fixes a b c :: "'a::ring_no_zero_divisors"
huffman@22990
   665
  shows "(a * c = b * c) = (c = 0 \<or> a = b)"
huffman@22990
   666
proof -
huffman@22990
   667
  have "(a * c = b * c) = ((a - b) * c = 0)"
nipkow@23477
   668
    by (simp add: ring_distribs)
huffman@22990
   669
  thus ?thesis
huffman@22990
   670
    by (simp add: disj_commute)
huffman@22990
   671
qed
paulson@14268
   672
paulson@24286
   673
lemma mult_cancel_left [simp,noatp]:
huffman@22990
   674
  fixes a b c :: "'a::ring_no_zero_divisors"
huffman@22990
   675
  shows "(c * a = c * b) = (c = 0 \<or> a = b)"
huffman@22990
   676
proof -
huffman@22990
   677
  have "(c * a = c * b) = (c * (a - b) = 0)"
nipkow@23477
   678
    by (simp add: ring_distribs)
huffman@22990
   679
  thus ?thesis
huffman@22990
   680
    by simp
huffman@22990
   681
qed
paulson@14268
   682
paulson@15234
   683
paulson@15234
   684
subsubsection{*Special Cancellation Simprules for Multiplication*}
paulson@15234
   685
paulson@15234
   686
text{*These also produce two cases when the comparison is a goal.*}
paulson@15234
   687
paulson@15234
   688
lemma mult_le_cancel_right1:
paulson@15234
   689
  fixes c :: "'a :: ordered_idom"
paulson@15234
   690
  shows "(c \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
paulson@15234
   691
by (insert mult_le_cancel_right [of 1 c b], simp)
paulson@15234
   692
paulson@15234
   693
lemma mult_le_cancel_right2:
paulson@15234
   694
  fixes c :: "'a :: ordered_idom"
paulson@15234
   695
  shows "(a*c \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
paulson@15234
   696
by (insert mult_le_cancel_right [of a c 1], simp)
paulson@15234
   697
paulson@15234
   698
lemma mult_le_cancel_left1:
paulson@15234
   699
  fixes c :: "'a :: ordered_idom"
paulson@15234
   700
  shows "(c \<le> c*b) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
paulson@15234
   701
by (insert mult_le_cancel_left [of c 1 b], simp)
paulson@15234
   702
paulson@15234
   703
lemma mult_le_cancel_left2:
paulson@15234
   704
  fixes c :: "'a :: ordered_idom"
paulson@15234
   705
  shows "(c*a \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
paulson@15234
   706
by (insert mult_le_cancel_left [of c a 1], simp)
paulson@15234
   707
paulson@15234
   708
lemma mult_less_cancel_right1:
paulson@15234
   709
  fixes c :: "'a :: ordered_idom"
paulson@15234
   710
  shows "(c < b*c) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
paulson@15234
   711
by (insert mult_less_cancel_right [of 1 c b], simp)
paulson@15234
   712
paulson@15234
   713
lemma mult_less_cancel_right2:
paulson@15234
   714
  fixes c :: "'a :: ordered_idom"
paulson@15234
   715
  shows "(a*c < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
paulson@15234
   716
by (insert mult_less_cancel_right [of a c 1], simp)
paulson@15234
   717
paulson@15234
   718
lemma mult_less_cancel_left1:
paulson@15234
   719
  fixes c :: "'a :: ordered_idom"
paulson@15234
   720
  shows "(c < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
paulson@15234
   721
by (insert mult_less_cancel_left [of c 1 b], simp)
paulson@15234
   722
paulson@15234
   723
lemma mult_less_cancel_left2:
paulson@15234
   724
  fixes c :: "'a :: ordered_idom"
paulson@15234
   725
  shows "(c*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
paulson@15234
   726
by (insert mult_less_cancel_left [of c a 1], simp)
paulson@15234
   727
paulson@15234
   728
lemma mult_cancel_right1 [simp]:
huffman@23544
   729
  fixes c :: "'a :: ring_1_no_zero_divisors"
paulson@15234
   730
  shows "(c = b*c) = (c = 0 | b=1)"
paulson@15234
   731
by (insert mult_cancel_right [of 1 c b], force)
paulson@15234
   732
paulson@15234
   733
lemma mult_cancel_right2 [simp]:
huffman@23544
   734
  fixes c :: "'a :: ring_1_no_zero_divisors"
paulson@15234
   735
  shows "(a*c = c) = (c = 0 | a=1)"
paulson@15234
   736
by (insert mult_cancel_right [of a c 1], simp)
paulson@15234
   737
 
paulson@15234
   738
lemma mult_cancel_left1 [simp]:
huffman@23544
   739
  fixes c :: "'a :: ring_1_no_zero_divisors"
paulson@15234
   740
  shows "(c = c*b) = (c = 0 | b=1)"
paulson@15234
   741
by (insert mult_cancel_left [of c 1 b], force)
paulson@15234
   742
paulson@15234
   743
lemma mult_cancel_left2 [simp]:
huffman@23544
   744
  fixes c :: "'a :: ring_1_no_zero_divisors"
paulson@15234
   745
  shows "(c*a = c) = (c = 0 | a=1)"
paulson@15234
   746
by (insert mult_cancel_left [of c a 1], simp)
paulson@15234
   747
paulson@15234
   748
paulson@15234
   749
text{*Simprules for comparisons where common factors can be cancelled.*}
paulson@15234
   750
lemmas mult_compare_simps =
paulson@15234
   751
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
   752
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
   753
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
   754
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
   755
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
   756
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
   757
    mult_cancel_right mult_cancel_left
paulson@15234
   758
    mult_cancel_right1 mult_cancel_right2
paulson@15234
   759
    mult_cancel_left1 mult_cancel_left2
paulson@15234
   760
paulson@15234
   761
paulson@14265
   762
subsection {* Fields *}
paulson@14265
   763
paulson@14288
   764
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
paulson@14288
   765
proof
paulson@14288
   766
  assume neq: "b \<noteq> 0"
paulson@14288
   767
  {
paulson@14288
   768
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
paulson@14288
   769
    also assume "a / b = 1"
paulson@14288
   770
    finally show "a = b" by simp
paulson@14288
   771
  next
paulson@14288
   772
    assume "a = b"
paulson@14288
   773
    with neq show "a / b = 1" by (simp add: divide_inverse)
paulson@14288
   774
  }
paulson@14288
   775
qed
paulson@14288
   776
paulson@14288
   777
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"
paulson@14288
   778
by (simp add: divide_inverse)
paulson@14288
   779
nipkow@23398
   780
lemma divide_self[simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
paulson@14288
   781
  by (simp add: divide_inverse)
paulson@14288
   782
paulson@14430
   783
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"
paulson@14430
   784
by (simp add: divide_inverse)
paulson@14277
   785
paulson@15228
   786
lemma divide_self_if [simp]:
paulson@15228
   787
     "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
paulson@15228
   788
  by (simp add: divide_self)
paulson@15228
   789
paulson@14430
   790
lemma divide_zero_left [simp]: "0/a = (0::'a::field)"
paulson@14430
   791
by (simp add: divide_inverse)
paulson@14277
   792
paulson@14430
   793
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"
paulson@14430
   794
by (simp add: divide_inverse)
paulson@14277
   795
paulson@14430
   796
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"
nipkow@23477
   797
by (simp add: divide_inverse ring_distribs) 
paulson@14293
   798
nipkow@23482
   799
(* what ordering?? this is a straight instance of mult_eq_0_iff
paulson@14270
   800
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
paulson@14270
   801
      of an ordering.*}
huffman@20496
   802
lemma field_mult_eq_0_iff [simp]:
huffman@20496
   803
  "(a*b = (0::'a::division_ring)) = (a = 0 | b = 0)"
huffman@22990
   804
by simp
nipkow@23482
   805
*)
nipkow@23496
   806
(* subsumed by mult_cancel lemmas on ring_no_zero_divisors
paulson@14268
   807
text{*Cancellation of equalities with a common factor*}
paulson@14268
   808
lemma field_mult_cancel_right_lemma:
huffman@20496
   809
      assumes cnz: "c \<noteq> (0::'a::division_ring)"
huffman@20496
   810
         and eq:  "a*c = b*c"
huffman@20496
   811
        shows "a=b"
paulson@14377
   812
proof -
paulson@14268
   813
  have "(a * c) * inverse c = (b * c) * inverse c"
paulson@14268
   814
    by (simp add: eq)
paulson@14268
   815
  thus "a=b"
paulson@14268
   816
    by (simp add: mult_assoc cnz)
paulson@14377
   817
qed
paulson@14268
   818
paulson@14348
   819
lemma field_mult_cancel_right [simp]:
huffman@20496
   820
     "(a*c = b*c) = (c = (0::'a::division_ring) | a=b)"
huffman@22990
   821
by simp
paulson@14268
   822
paulson@14348
   823
lemma field_mult_cancel_left [simp]:
huffman@20496
   824
     "(c*a = c*b) = (c = (0::'a::division_ring) | a=b)"
huffman@22990
   825
by simp
nipkow@23496
   826
*)
huffman@20496
   827
lemma nonzero_imp_inverse_nonzero:
huffman@20496
   828
  "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::division_ring)"
paulson@14377
   829
proof
paulson@14268
   830
  assume ianz: "inverse a = 0"
paulson@14268
   831
  assume "a \<noteq> 0"
paulson@14268
   832
  hence "1 = a * inverse a" by simp
paulson@14268
   833
  also have "... = 0" by (simp add: ianz)
huffman@20496
   834
  finally have "1 = (0::'a::division_ring)" .
paulson@14268
   835
  thus False by (simp add: eq_commute)
paulson@14377
   836
qed
paulson@14268
   837
paulson@14277
   838
paulson@14277
   839
subsection{*Basic Properties of @{term inverse}*}
paulson@14277
   840
huffman@20496
   841
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::division_ring)"
paulson@14268
   842
apply (rule ccontr) 
paulson@14268
   843
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   844
done
paulson@14268
   845
paulson@14268
   846
lemma inverse_nonzero_imp_nonzero:
huffman@20496
   847
   "inverse a = 0 ==> a = (0::'a::division_ring)"
paulson@14268
   848
apply (rule ccontr) 
paulson@14268
   849
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   850
done
paulson@14268
   851
paulson@14268
   852
lemma inverse_nonzero_iff_nonzero [simp]:
huffman@20496
   853
   "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
paulson@14268
   854
by (force dest: inverse_nonzero_imp_nonzero) 
paulson@14268
   855
paulson@14268
   856
lemma nonzero_inverse_minus_eq:
huffman@20496
   857
      assumes [simp]: "a\<noteq>0"
huffman@20496
   858
      shows "inverse(-a) = -inverse(a::'a::division_ring)"
paulson@14377
   859
proof -
paulson@14377
   860
  have "-a * inverse (- a) = -a * - inverse a"
paulson@14377
   861
    by simp
paulson@14377
   862
  thus ?thesis 
nipkow@23496
   863
    by (simp only: mult_cancel_left, simp)
paulson@14377
   864
qed
paulson@14268
   865
paulson@14268
   866
lemma inverse_minus_eq [simp]:
huffman@20496
   867
   "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
paulson@14377
   868
proof cases
paulson@14377
   869
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
paulson@14377
   870
next
paulson@14377
   871
  assume "a\<noteq>0" 
paulson@14377
   872
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
paulson@14377
   873
qed
paulson@14268
   874
paulson@14268
   875
lemma nonzero_inverse_eq_imp_eq:
paulson@14269
   876
      assumes inveq: "inverse a = inverse b"
paulson@14269
   877
	  and anz:  "a \<noteq> 0"
paulson@14269
   878
	  and bnz:  "b \<noteq> 0"
huffman@20496
   879
	 shows "a = (b::'a::division_ring)"
paulson@14377
   880
proof -
paulson@14268
   881
  have "a * inverse b = a * inverse a"
paulson@14268
   882
    by (simp add: inveq)
paulson@14268
   883
  hence "(a * inverse b) * b = (a * inverse a) * b"
paulson@14268
   884
    by simp
paulson@14268
   885
  thus "a = b"
paulson@14268
   886
    by (simp add: mult_assoc anz bnz)
paulson@14377
   887
qed
paulson@14268
   888
paulson@14268
   889
lemma inverse_eq_imp_eq:
huffman@20496
   890
  "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
haftmann@21328
   891
apply (cases "a=0 | b=0") 
paulson@14268
   892
 apply (force dest!: inverse_zero_imp_zero
paulson@14268
   893
              simp add: eq_commute [of "0::'a"])
paulson@14268
   894
apply (force dest!: nonzero_inverse_eq_imp_eq) 
paulson@14268
   895
done
paulson@14268
   896
paulson@14268
   897
lemma inverse_eq_iff_eq [simp]:
huffman@20496
   898
  "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
huffman@20496
   899
by (force dest!: inverse_eq_imp_eq)
paulson@14268
   900
paulson@14270
   901
lemma nonzero_inverse_inverse_eq:
huffman@20496
   902
      assumes [simp]: "a \<noteq> 0"
huffman@20496
   903
      shows "inverse(inverse (a::'a::division_ring)) = a"
paulson@14270
   904
  proof -
paulson@14270
   905
  have "(inverse (inverse a) * inverse a) * a = a" 
paulson@14270
   906
    by (simp add: nonzero_imp_inverse_nonzero)
paulson@14270
   907
  thus ?thesis
paulson@14270
   908
    by (simp add: mult_assoc)
paulson@14270
   909
  qed
paulson@14270
   910
paulson@14270
   911
lemma inverse_inverse_eq [simp]:
huffman@20496
   912
     "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
paulson@14270
   913
  proof cases
paulson@14270
   914
    assume "a=0" thus ?thesis by simp
paulson@14270
   915
  next
paulson@14270
   916
    assume "a\<noteq>0" 
paulson@14270
   917
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
paulson@14270
   918
  qed
paulson@14270
   919
huffman@20496
   920
lemma inverse_1 [simp]: "inverse 1 = (1::'a::division_ring)"
paulson@14270
   921
  proof -
huffman@20496
   922
  have "inverse 1 * 1 = (1::'a::division_ring)" 
paulson@14270
   923
    by (rule left_inverse [OF zero_neq_one [symmetric]])
paulson@14270
   924
  thus ?thesis  by simp
paulson@14270
   925
  qed
paulson@14270
   926
paulson@15077
   927
lemma inverse_unique: 
paulson@15077
   928
  assumes ab: "a*b = 1"
huffman@20496
   929
  shows "inverse a = (b::'a::division_ring)"
paulson@15077
   930
proof -
paulson@15077
   931
  have "a \<noteq> 0" using ab by auto
paulson@15077
   932
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
paulson@15077
   933
  ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
paulson@15077
   934
qed
paulson@15077
   935
paulson@14270
   936
lemma nonzero_inverse_mult_distrib: 
paulson@14270
   937
      assumes anz: "a \<noteq> 0"
paulson@14270
   938
          and bnz: "b \<noteq> 0"
huffman@20496
   939
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::division_ring)"
paulson@14270
   940
  proof -
paulson@14270
   941
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
nipkow@23482
   942
    by (simp add: anz bnz)
paulson@14270
   943
  hence "inverse(a*b) * a = inverse(b)" 
paulson@14270
   944
    by (simp add: mult_assoc bnz)
paulson@14270
   945
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
paulson@14270
   946
    by simp
paulson@14270
   947
  thus ?thesis
paulson@14270
   948
    by (simp add: mult_assoc anz)
paulson@14270
   949
  qed
paulson@14270
   950
paulson@14270
   951
text{*This version builds in division by zero while also re-orienting
paulson@14270
   952
      the right-hand side.*}
paulson@14270
   953
lemma inverse_mult_distrib [simp]:
paulson@14270
   954
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
paulson@14270
   955
  proof cases
paulson@14270
   956
    assume "a \<noteq> 0 & b \<noteq> 0" 
haftmann@22993
   957
    thus ?thesis
haftmann@22993
   958
      by (simp add: nonzero_inverse_mult_distrib mult_commute)
paulson@14270
   959
  next
paulson@14270
   960
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
haftmann@22993
   961
    thus ?thesis
haftmann@22993
   962
      by force
paulson@14270
   963
  qed
paulson@14270
   964
huffman@20496
   965
lemma division_ring_inverse_add:
huffman@20496
   966
  "[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
huffman@20496
   967
   ==> inverse a + inverse b = inverse a * (a+b) * inverse b"
nipkow@23477
   968
by (simp add: ring_simps)
huffman@20496
   969
huffman@20496
   970
lemma division_ring_inverse_diff:
huffman@20496
   971
  "[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
huffman@20496
   972
   ==> inverse a - inverse b = inverse a * (b-a) * inverse b"
nipkow@23477
   973
by (simp add: ring_simps)
huffman@20496
   974
paulson@14270
   975
text{*There is no slick version using division by zero.*}
paulson@14270
   976
lemma inverse_add:
nipkow@23477
   977
  "[|a \<noteq> 0;  b \<noteq> 0|]
nipkow@23477
   978
   ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
huffman@20496
   979
by (simp add: division_ring_inverse_add mult_ac)
paulson@14270
   980
paulson@14365
   981
lemma inverse_divide [simp]:
nipkow@23477
   982
  "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
nipkow@23477
   983
by (simp add: divide_inverse mult_commute)
paulson@14365
   984
wenzelm@23389
   985
avigad@16775
   986
subsection {* Calculations with fractions *}
avigad@16775
   987
nipkow@23413
   988
text{* There is a whole bunch of simp-rules just for class @{text
nipkow@23413
   989
field} but none for class @{text field} and @{text nonzero_divides}
nipkow@23413
   990
because the latter are covered by a simproc. *}
nipkow@23413
   991
nipkow@23413
   992
lemma nonzero_mult_divide_mult_cancel_left[simp]:
nipkow@23477
   993
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/(b::'a::field)"
paulson@14277
   994
proof -
paulson@14277
   995
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
nipkow@23482
   996
    by (simp add: divide_inverse nonzero_inverse_mult_distrib)
paulson@14277
   997
  also have "... =  a * inverse b * (inverse c * c)"
paulson@14277
   998
    by (simp only: mult_ac)
paulson@14277
   999
  also have "... =  a * inverse b"
paulson@14277
  1000
    by simp
paulson@14277
  1001
    finally show ?thesis 
paulson@14277
  1002
    by (simp add: divide_inverse)
paulson@14277
  1003
qed
paulson@14277
  1004
nipkow@23413
  1005
lemma mult_divide_mult_cancel_left:
nipkow@23477
  1006
  "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1007
apply (cases "b = 0")
nipkow@23413
  1008
apply (simp_all add: nonzero_mult_divide_mult_cancel_left)
paulson@14277
  1009
done
paulson@14277
  1010
nipkow@23413
  1011
lemma nonzero_mult_divide_mult_cancel_right:
nipkow@23477
  1012
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
nipkow@23413
  1013
by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) 
paulson@14321
  1014
nipkow@23413
  1015
lemma mult_divide_mult_cancel_right:
nipkow@23477
  1016
  "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1017
apply (cases "b = 0")
nipkow@23413
  1018
apply (simp_all add: nonzero_mult_divide_mult_cancel_right)
paulson@14321
  1019
done
nipkow@23413
  1020
paulson@14284
  1021
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
nipkow@23477
  1022
by (simp add: divide_inverse)
paulson@14284
  1023
paulson@15234
  1024
lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)"
paulson@14430
  1025
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1026
paulson@14430
  1027
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
paulson@14430
  1028
by (simp add: divide_inverse mult_ac)
paulson@14288
  1029
nipkow@23482
  1030
lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left
nipkow@23482
  1031
paulson@24286
  1032
lemma divide_divide_eq_right [simp,noatp]:
nipkow@23477
  1033
  "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
paulson@14430
  1034
by (simp add: divide_inverse mult_ac)
paulson@14288
  1035
paulson@24286
  1036
lemma divide_divide_eq_left [simp,noatp]:
nipkow@23477
  1037
  "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
paulson@14430
  1038
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1039
avigad@16775
  1040
lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1041
    x / y + w / z = (x * z + w * y) / (y * z)"
nipkow@23477
  1042
apply (subgoal_tac "x / y = (x * z) / (y * z)")
nipkow@23477
  1043
apply (erule ssubst)
nipkow@23477
  1044
apply (subgoal_tac "w / z = (w * y) / (y * z)")
nipkow@23477
  1045
apply (erule ssubst)
nipkow@23477
  1046
apply (rule add_divide_distrib [THEN sym])
nipkow@23477
  1047
apply (subst mult_commute)
nipkow@23477
  1048
apply (erule nonzero_mult_divide_mult_cancel_left [THEN sym])
nipkow@23477
  1049
apply assumption
nipkow@23477
  1050
apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym])
nipkow@23477
  1051
apply assumption
avigad@16775
  1052
done
paulson@14268
  1053
wenzelm@23389
  1054
paulson@15234
  1055
subsubsection{*Special Cancellation Simprules for Division*}
paulson@15234
  1056
nipkow@23413
  1057
lemma mult_divide_mult_cancel_left_if[simp]:
nipkow@23477
  1058
fixes c :: "'a :: {field,division_by_zero}"
nipkow@23477
  1059
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
nipkow@23413
  1060
by (simp add: mult_divide_mult_cancel_left)
nipkow@23413
  1061
nipkow@23413
  1062
lemma nonzero_mult_divide_cancel_right[simp]:
nipkow@23413
  1063
  "b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
nipkow@23413
  1064
using nonzero_mult_divide_mult_cancel_right[of 1 b a] by simp
nipkow@23413
  1065
nipkow@23413
  1066
lemma nonzero_mult_divide_cancel_left[simp]:
nipkow@23413
  1067
  "a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)"
nipkow@23413
  1068
using nonzero_mult_divide_mult_cancel_left[of 1 a b] by simp
nipkow@23413
  1069
nipkow@23413
  1070
nipkow@23413
  1071
lemma nonzero_divide_mult_cancel_right[simp]:
nipkow@23413
  1072
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> b / (a * b) = 1/(a::'a::field)"
nipkow@23413
  1073
using nonzero_mult_divide_mult_cancel_right[of a b 1] by simp
nipkow@23413
  1074
nipkow@23413
  1075
lemma nonzero_divide_mult_cancel_left[simp]:
nipkow@23413
  1076
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> a / (a * b) = 1/(b::'a::field)"
nipkow@23413
  1077
using nonzero_mult_divide_mult_cancel_left[of b a 1] by simp
nipkow@23413
  1078
nipkow@23413
  1079
nipkow@23413
  1080
lemma nonzero_mult_divide_mult_cancel_left2[simp]:
nipkow@23477
  1081
  "[|b\<noteq>0; c\<noteq>0|] ==> (c*a) / (b*c) = a/(b::'a::field)"
nipkow@23413
  1082
using nonzero_mult_divide_mult_cancel_left[of b c a] by(simp add:mult_ac)
nipkow@23413
  1083
nipkow@23413
  1084
lemma nonzero_mult_divide_mult_cancel_right2[simp]:
nipkow@23477
  1085
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (c*b) = a/(b::'a::field)"
nipkow@23413
  1086
using nonzero_mult_divide_mult_cancel_right[of b c a] by(simp add:mult_ac)
nipkow@23413
  1087
paulson@15234
  1088
paulson@14293
  1089
subsection {* Division and Unary Minus *}
paulson@14293
  1090
paulson@14293
  1091
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
paulson@14293
  1092
by (simp add: divide_inverse minus_mult_left)
paulson@14293
  1093
paulson@14293
  1094
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
paulson@14293
  1095
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
paulson@14293
  1096
paulson@14293
  1097
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
paulson@14293
  1098
by (simp add: divide_inverse nonzero_inverse_minus_eq)
paulson@14293
  1099
paulson@14430
  1100
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
paulson@14430
  1101
by (simp add: divide_inverse minus_mult_left [symmetric])
paulson@14293
  1102
paulson@14293
  1103
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
paulson@14430
  1104
by (simp add: divide_inverse minus_mult_right [symmetric])
paulson@14430
  1105
paulson@14293
  1106
paulson@14293
  1107
text{*The effect is to extract signs from divisions*}
paulson@17085
  1108
lemmas divide_minus_left = minus_divide_left [symmetric]
paulson@17085
  1109
lemmas divide_minus_right = minus_divide_right [symmetric]
paulson@17085
  1110
declare divide_minus_left [simp]   divide_minus_right [simp]
paulson@14293
  1111
paulson@14387
  1112
text{*Also, extract signs from products*}
paulson@17085
  1113
lemmas mult_minus_left = minus_mult_left [symmetric]
paulson@17085
  1114
lemmas mult_minus_right = minus_mult_right [symmetric]
paulson@17085
  1115
declare mult_minus_left [simp]   mult_minus_right [simp]
paulson@14387
  1116
paulson@14293
  1117
lemma minus_divide_divide [simp]:
nipkow@23477
  1118
  "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1119
apply (cases "b=0", simp) 
paulson@14293
  1120
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
  1121
done
paulson@14293
  1122
paulson@14430
  1123
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
paulson@14387
  1124
by (simp add: diff_minus add_divide_distrib) 
paulson@14387
  1125
nipkow@23482
  1126
lemma add_divide_eq_iff:
nipkow@23482
  1127
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z"
nipkow@23482
  1128
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1129
nipkow@23482
  1130
lemma divide_add_eq_iff:
nipkow@23482
  1131
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z"
nipkow@23482
  1132
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1133
nipkow@23482
  1134
lemma diff_divide_eq_iff:
nipkow@23482
  1135
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x - y/z = (z*x - y)/z"
nipkow@23482
  1136
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1137
nipkow@23482
  1138
lemma divide_diff_eq_iff:
nipkow@23482
  1139
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z - y = (x - z*y)/z"
nipkow@23482
  1140
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1141
nipkow@23482
  1142
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
nipkow@23482
  1143
proof -
nipkow@23482
  1144
  assume [simp]: "c\<noteq>0"
nipkow@23496
  1145
  have "(a = b/c) = (a*c = (b/c)*c)" by simp
nipkow@23496
  1146
  also have "... = (a*c = b)" by (simp add: divide_inverse mult_assoc)
nipkow@23482
  1147
  finally show ?thesis .
nipkow@23482
  1148
qed
nipkow@23482
  1149
nipkow@23482
  1150
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
nipkow@23482
  1151
proof -
nipkow@23482
  1152
  assume [simp]: "c\<noteq>0"
nipkow@23496
  1153
  have "(b/c = a) = ((b/c)*c = a*c)"  by simp
nipkow@23496
  1154
  also have "... = (b = a*c)"  by (simp add: divide_inverse mult_assoc) 
nipkow@23482
  1155
  finally show ?thesis .
nipkow@23482
  1156
qed
nipkow@23482
  1157
nipkow@23482
  1158
lemma eq_divide_eq:
nipkow@23482
  1159
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
nipkow@23482
  1160
by (simp add: nonzero_eq_divide_eq) 
nipkow@23482
  1161
nipkow@23482
  1162
lemma divide_eq_eq:
nipkow@23482
  1163
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
nipkow@23482
  1164
by (force simp add: nonzero_divide_eq_eq) 
nipkow@23482
  1165
nipkow@23482
  1166
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
nipkow@23482
  1167
    b = a * c ==> b / c = a"
nipkow@23482
  1168
  by (subst divide_eq_eq, simp)
nipkow@23482
  1169
nipkow@23482
  1170
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
nipkow@23482
  1171
    a * c = b ==> a = b / c"
nipkow@23482
  1172
  by (subst eq_divide_eq, simp)
nipkow@23482
  1173
nipkow@23482
  1174
nipkow@23482
  1175
lemmas field_eq_simps = ring_simps
nipkow@23482
  1176
  (* pull / out*)
nipkow@23482
  1177
  add_divide_eq_iff divide_add_eq_iff
nipkow@23482
  1178
  diff_divide_eq_iff divide_diff_eq_iff
nipkow@23482
  1179
  (* multiply eqn *)
nipkow@23482
  1180
  nonzero_eq_divide_eq nonzero_divide_eq_eq
nipkow@23482
  1181
(* is added later:
nipkow@23482
  1182
  times_divide_eq_left times_divide_eq_right
nipkow@23482
  1183
*)
nipkow@23482
  1184
nipkow@23482
  1185
text{*An example:*}
nipkow@23482
  1186
lemma fixes a b c d e f :: "'a::field"
nipkow@23482
  1187
shows "\<lbrakk>a\<noteq>b; c\<noteq>d; e\<noteq>f \<rbrakk> \<Longrightarrow> ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) = 1"
nipkow@23482
  1188
apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")
nipkow@23482
  1189
 apply(simp add:field_eq_simps)
nipkow@23482
  1190
apply(simp)
nipkow@23482
  1191
done
nipkow@23482
  1192
nipkow@23482
  1193
avigad@16775
  1194
lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1195
    x / y - w / z = (x * z - w * y) / (y * z)"
nipkow@23482
  1196
by (simp add:field_eq_simps times_divide_eq)
nipkow@23482
  1197
nipkow@23482
  1198
lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
nipkow@23482
  1199
    (x / y = w / z) = (x * z = w * y)"
nipkow@23482
  1200
by (simp add:field_eq_simps times_divide_eq)
paulson@14293
  1201
wenzelm@23389
  1202
paulson@14268
  1203
subsection {* Ordered Fields *}
paulson@14268
  1204
paulson@14277
  1205
lemma positive_imp_inverse_positive: 
nipkow@23482
  1206
assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
nipkow@23482
  1207
proof -
paulson@14268
  1208
  have "0 < a * inverse a" 
paulson@14268
  1209
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
paulson@14268
  1210
  thus "0 < inverse a" 
paulson@14268
  1211
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
nipkow@23482
  1212
qed
paulson@14268
  1213
paulson@14277
  1214
lemma negative_imp_inverse_negative:
nipkow@23482
  1215
  "a < 0 ==> inverse a < (0::'a::ordered_field)"
nipkow@23482
  1216
by (insert positive_imp_inverse_positive [of "-a"], 
nipkow@23482
  1217
    simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)
paulson@14268
  1218
paulson@14268
  1219
lemma inverse_le_imp_le:
nipkow@23482
  1220
assumes invle: "inverse a \<le> inverse b" and apos:  "0 < a"
nipkow@23482
  1221
shows "b \<le> (a::'a::ordered_field)"
nipkow@23482
  1222
proof (rule classical)
paulson@14268
  1223
  assume "~ b \<le> a"
nipkow@23482
  1224
  hence "a < b"  by (simp add: linorder_not_le)
nipkow@23482
  1225
  hence bpos: "0 < b"  by (blast intro: apos order_less_trans)
paulson@14268
  1226
  hence "a * inverse a \<le> a * inverse b"
paulson@14268
  1227
    by (simp add: apos invle order_less_imp_le mult_left_mono)
paulson@14268
  1228
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
paulson@14268
  1229
    by (simp add: bpos order_less_imp_le mult_right_mono)
nipkow@23482
  1230
  thus "b \<le> a"  by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
nipkow@23482
  1231
qed
paulson@14268
  1232
paulson@14277
  1233
lemma inverse_positive_imp_positive:
nipkow@23482
  1234
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
nipkow@23482
  1235
shows "0 < (a::'a::ordered_field)"
wenzelm@23389
  1236
proof -
paulson@14277
  1237
  have "0 < inverse (inverse a)"
wenzelm@23389
  1238
    using inv_gt_0 by (rule positive_imp_inverse_positive)
paulson@14277
  1239
  thus "0 < a"
wenzelm@23389
  1240
    using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1241
qed
paulson@14277
  1242
paulson@14277
  1243
lemma inverse_positive_iff_positive [simp]:
nipkow@23482
  1244
  "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1245
apply (cases "a = 0", simp)
paulson@14277
  1246
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
  1247
done
paulson@14277
  1248
paulson@14277
  1249
lemma inverse_negative_imp_negative:
nipkow@23482
  1250
assumes inv_less_0: "inverse a < 0" and nz:  "a \<noteq> 0"
nipkow@23482
  1251
shows "a < (0::'a::ordered_field)"
wenzelm@23389
  1252
proof -
paulson@14277
  1253
  have "inverse (inverse a) < 0"
wenzelm@23389
  1254
    using inv_less_0 by (rule negative_imp_inverse_negative)
nipkow@23482
  1255
  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1256
qed
paulson@14277
  1257
paulson@14277
  1258
lemma inverse_negative_iff_negative [simp]:
nipkow@23482
  1259
  "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1260
apply (cases "a = 0", simp)
paulson@14277
  1261
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
  1262
done
paulson@14277
  1263
paulson@14277
  1264
lemma inverse_nonnegative_iff_nonnegative [simp]:
nipkow@23482
  1265
  "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1266
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1267
paulson@14277
  1268
lemma inverse_nonpositive_iff_nonpositive [simp]:
nipkow@23482
  1269
  "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1270
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1271
chaieb@23406
  1272
lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)"
chaieb@23406
  1273
proof
chaieb@23406
  1274
  fix x::'a
chaieb@23406
  1275
  have m1: "- (1::'a) < 0" by simp
chaieb@23406
  1276
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1277
  have "(- 1) + x < x" by simp
chaieb@23406
  1278
  thus "\<exists>y. y < x" by blast
chaieb@23406
  1279
qed
chaieb@23406
  1280
chaieb@23406
  1281
lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)"
chaieb@23406
  1282
proof
chaieb@23406
  1283
  fix x::'a
chaieb@23406
  1284
  have m1: " (1::'a) > 0" by simp
chaieb@23406
  1285
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1286
  have "1 + x > x" by simp
chaieb@23406
  1287
  thus "\<exists>y. y > x" by blast
chaieb@23406
  1288
qed
paulson@14277
  1289
paulson@14277
  1290
subsection{*Anti-Monotonicity of @{term inverse}*}
paulson@14277
  1291
paulson@14268
  1292
lemma less_imp_inverse_less:
nipkow@23482
  1293
assumes less: "a < b" and apos:  "0 < a"
nipkow@23482
  1294
shows "inverse b < inverse (a::'a::ordered_field)"
nipkow@23482
  1295
proof (rule ccontr)
paulson@14268
  1296
  assume "~ inverse b < inverse a"
paulson@14268
  1297
  hence "inverse a \<le> inverse b"
paulson@14268
  1298
    by (simp add: linorder_not_less)
paulson@14268
  1299
  hence "~ (a < b)"
paulson@14268
  1300
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
paulson@14268
  1301
  thus False
paulson@14268
  1302
    by (rule notE [OF _ less])
nipkow@23482
  1303
qed
paulson@14268
  1304
paulson@14268
  1305
lemma inverse_less_imp_less:
nipkow@23482
  1306
  "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
paulson@14268
  1307
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
paulson@14268
  1308
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
paulson@14268
  1309
done
paulson@14268
  1310
paulson@14268
  1311
text{*Both premises are essential. Consider -1 and 1.*}
paulson@24286
  1312
lemma inverse_less_iff_less [simp,noatp]:
nipkow@23482
  1313
  "[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1314
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
paulson@14268
  1315
paulson@14268
  1316
lemma le_imp_inverse_le:
nipkow@23482
  1317
  "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
nipkow@23482
  1318
by (force simp add: order_le_less less_imp_inverse_less)
paulson@14268
  1319
paulson@24286
  1320
lemma inverse_le_iff_le [simp,noatp]:
nipkow@23482
  1321
 "[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1322
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
paulson@14268
  1323
paulson@14268
  1324
paulson@14268
  1325
text{*These results refer to both operands being negative.  The opposite-sign
paulson@14268
  1326
case is trivial, since inverse preserves signs.*}
paulson@14268
  1327
lemma inverse_le_imp_le_neg:
nipkow@23482
  1328
  "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
nipkow@23482
  1329
apply (rule classical) 
nipkow@23482
  1330
apply (subgoal_tac "a < 0") 
nipkow@23482
  1331
 prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
nipkow@23482
  1332
apply (insert inverse_le_imp_le [of "-b" "-a"])
nipkow@23482
  1333
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1334
done
paulson@14268
  1335
paulson@14268
  1336
lemma less_imp_inverse_less_neg:
paulson@14268
  1337
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
nipkow@23482
  1338
apply (subgoal_tac "a < 0") 
nipkow@23482
  1339
 prefer 2 apply (blast intro: order_less_trans) 
nipkow@23482
  1340
apply (insert less_imp_inverse_less [of "-b" "-a"])
nipkow@23482
  1341
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1342
done
paulson@14268
  1343
paulson@14268
  1344
lemma inverse_less_imp_less_neg:
paulson@14268
  1345
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
nipkow@23482
  1346
apply (rule classical) 
nipkow@23482
  1347
apply (subgoal_tac "a < 0") 
nipkow@23482
  1348
 prefer 2
nipkow@23482
  1349
 apply (force simp add: linorder_not_less intro: order_le_less_trans) 
nipkow@23482
  1350
apply (insert inverse_less_imp_less [of "-b" "-a"])
nipkow@23482
  1351
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1352
done
paulson@14268
  1353
paulson@24286
  1354
lemma inverse_less_iff_less_neg [simp,noatp]:
nipkow@23482
  1355
  "[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
nipkow@23482
  1356
apply (insert inverse_less_iff_less [of "-b" "-a"])
nipkow@23482
  1357
apply (simp del: inverse_less_iff_less 
nipkow@23482
  1358
            add: order_less_imp_not_eq nonzero_inverse_minus_eq)
nipkow@23482
  1359
done
paulson@14268
  1360
paulson@14268
  1361
lemma le_imp_inverse_le_neg:
nipkow@23482
  1362
  "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
nipkow@23482
  1363
by (force simp add: order_le_less less_imp_inverse_less_neg)
paulson@14268
  1364
paulson@24286
  1365
lemma inverse_le_iff_le_neg [simp,noatp]:
nipkow@23482
  1366
 "[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1367
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
paulson@14265
  1368
paulson@14277
  1369
paulson@14365
  1370
subsection{*Inverses and the Number One*}
paulson@14365
  1371
paulson@14365
  1372
lemma one_less_inverse_iff:
nipkow@23482
  1373
  "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"
nipkow@23482
  1374
proof cases
paulson@14365
  1375
  assume "0 < x"
paulson@14365
  1376
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
  1377
    show ?thesis by simp
paulson@14365
  1378
next
paulson@14365
  1379
  assume notless: "~ (0 < x)"
paulson@14365
  1380
  have "~ (1 < inverse x)"
paulson@14365
  1381
  proof
paulson@14365
  1382
    assume "1 < inverse x"
paulson@14365
  1383
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
paulson@14365
  1384
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
  1385
    finally show False by auto
paulson@14365
  1386
  qed
paulson@14365
  1387
  with notless show ?thesis by simp
paulson@14365
  1388
qed
paulson@14365
  1389
paulson@14365
  1390
lemma inverse_eq_1_iff [simp]:
nipkow@23482
  1391
  "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
paulson@14365
  1392
by (insert inverse_eq_iff_eq [of x 1], simp) 
paulson@14365
  1393
paulson@14365
  1394
lemma one_le_inverse_iff:
nipkow@23482
  1395
  "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1396
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
paulson@14365
  1397
                    eq_commute [of 1]) 
paulson@14365
  1398
paulson@14365
  1399
lemma inverse_less_1_iff:
nipkow@23482
  1400
  "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1401
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
paulson@14365
  1402
paulson@14365
  1403
lemma inverse_le_1_iff:
nipkow@23482
  1404
  "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1405
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
paulson@14365
  1406
wenzelm@23389
  1407
paulson@14288
  1408
subsection{*Simplification of Inequalities Involving Literal Divisors*}
paulson@14288
  1409
paulson@14288
  1410
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
paulson@14288
  1411
proof -
paulson@14288
  1412
  assume less: "0<c"
paulson@14288
  1413
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
paulson@14288
  1414
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1415
  also have "... = (a*c \<le> b)"
paulson@14288
  1416
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1417
  finally show ?thesis .
paulson@14288
  1418
qed
paulson@14288
  1419
paulson@14288
  1420
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
paulson@14288
  1421
proof -
paulson@14288
  1422
  assume less: "c<0"
paulson@14288
  1423
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
paulson@14288
  1424
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1425
  also have "... = (b \<le> a*c)"
paulson@14288
  1426
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1427
  finally show ?thesis .
paulson@14288
  1428
qed
paulson@14288
  1429
paulson@14288
  1430
lemma le_divide_eq:
paulson@14288
  1431
  "(a \<le> b/c) = 
paulson@14288
  1432
   (if 0 < c then a*c \<le> b
paulson@14288
  1433
             else if c < 0 then b \<le> a*c
paulson@14288
  1434
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1435
apply (cases "c=0", simp) 
paulson@14288
  1436
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
paulson@14288
  1437
done
paulson@14288
  1438
paulson@14288
  1439
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
paulson@14288
  1440
proof -
paulson@14288
  1441
  assume less: "0<c"
paulson@14288
  1442
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
paulson@14288
  1443
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1444
  also have "... = (b \<le> a*c)"
paulson@14288
  1445
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1446
  finally show ?thesis .
paulson@14288
  1447
qed
paulson@14288
  1448
paulson@14288
  1449
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
paulson@14288
  1450
proof -
paulson@14288
  1451
  assume less: "c<0"
paulson@14288
  1452
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
paulson@14288
  1453
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1454
  also have "... = (a*c \<le> b)"
paulson@14288
  1455
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1456
  finally show ?thesis .
paulson@14288
  1457
qed
paulson@14288
  1458
paulson@14288
  1459
lemma divide_le_eq:
paulson@14288
  1460
  "(b/c \<le> a) = 
paulson@14288
  1461
   (if 0 < c then b \<le> a*c
paulson@14288
  1462
             else if c < 0 then a*c \<le> b
paulson@14288
  1463
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1464
apply (cases "c=0", simp) 
paulson@14288
  1465
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
paulson@14288
  1466
done
paulson@14288
  1467
paulson@14288
  1468
lemma pos_less_divide_eq:
paulson@14288
  1469
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
paulson@14288
  1470
proof -
paulson@14288
  1471
  assume less: "0<c"
paulson@14288
  1472
  hence "(a < b/c) = (a*c < (b/c)*c)"
paulson@15234
  1473
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1474
  also have "... = (a*c < b)"
paulson@14288
  1475
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1476
  finally show ?thesis .
paulson@14288
  1477
qed
paulson@14288
  1478
paulson@14288
  1479
lemma neg_less_divide_eq:
paulson@14288
  1480
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
paulson@14288
  1481
proof -
paulson@14288
  1482
  assume less: "c<0"
paulson@14288
  1483
  hence "(a < b/c) = ((b/c)*c < a*c)"
paulson@15234
  1484
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1485
  also have "... = (b < a*c)"
paulson@14288
  1486
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1487
  finally show ?thesis .
paulson@14288
  1488
qed
paulson@14288
  1489
paulson@14288
  1490
lemma less_divide_eq:
paulson@14288
  1491
  "(a < b/c) = 
paulson@14288
  1492
   (if 0 < c then a*c < b
paulson@14288
  1493
             else if c < 0 then b < a*c
paulson@14288
  1494
             else  a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1495
apply (cases "c=0", simp) 
paulson@14288
  1496
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
paulson@14288
  1497
done
paulson@14288
  1498
paulson@14288
  1499
lemma pos_divide_less_eq:
paulson@14288
  1500
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
paulson@14288
  1501
proof -
paulson@14288
  1502
  assume less: "0<c"
paulson@14288
  1503
  hence "(b/c < a) = ((b/c)*c < a*c)"
paulson@15234
  1504
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1505
  also have "... = (b < a*c)"
paulson@14288
  1506
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1507
  finally show ?thesis .
paulson@14288
  1508
qed
paulson@14288
  1509
paulson@14288
  1510
lemma neg_divide_less_eq:
paulson@14288
  1511
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
paulson@14288
  1512
proof -
paulson@14288
  1513
  assume less: "c<0"
paulson@14288
  1514
  hence "(b/c < a) = (a*c < (b/c)*c)"
paulson@15234
  1515
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1516
  also have "... = (a*c < b)"
paulson@14288
  1517
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1518
  finally show ?thesis .
paulson@14288
  1519
qed
paulson@14288
  1520
paulson@14288
  1521
lemma divide_less_eq:
paulson@14288
  1522
  "(b/c < a) = 
paulson@14288
  1523
   (if 0 < c then b < a*c
paulson@14288
  1524
             else if c < 0 then a*c < b
paulson@14288
  1525
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1526
apply (cases "c=0", simp) 
paulson@14288
  1527
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
paulson@14288
  1528
done
paulson@14288
  1529
nipkow@23482
  1530
nipkow@23482
  1531
subsection{*Field simplification*}
nipkow@23482
  1532
nipkow@23482
  1533
text{* Lemmas @{text field_simps} multiply with denominators in
nipkow@23482
  1534
in(equations) if they can be proved to be non-zero (for equations) or
nipkow@23482
  1535
positive/negative (for inequations). *}
paulson@14288
  1536
nipkow@23482
  1537
lemmas field_simps = field_eq_simps
nipkow@23482
  1538
  (* multiply ineqn *)
nipkow@23482
  1539
  pos_divide_less_eq neg_divide_less_eq
nipkow@23482
  1540
  pos_less_divide_eq neg_less_divide_eq
nipkow@23482
  1541
  pos_divide_le_eq neg_divide_le_eq
nipkow@23482
  1542
  pos_le_divide_eq neg_le_divide_eq
paulson@14288
  1543
nipkow@23482
  1544
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
nipkow@23483
  1545
of positivity/negativity needed for @{text field_simps}. Have not added @{text
nipkow@23482
  1546
sign_simps} to @{text field_simps} because the former can lead to case
nipkow@23482
  1547
explosions. *}
paulson@14288
  1548
nipkow@23482
  1549
lemmas sign_simps = group_simps
nipkow@23482
  1550
  zero_less_mult_iff  mult_less_0_iff
paulson@14288
  1551
nipkow@23482
  1552
(* Only works once linear arithmetic is installed:
nipkow@23482
  1553
text{*An example:*}
nipkow@23482
  1554
lemma fixes a b c d e f :: "'a::ordered_field"
nipkow@23482
  1555
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
nipkow@23482
  1556
 ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
nipkow@23482
  1557
 ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
nipkow@23482
  1558
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
nipkow@23482
  1559
 prefer 2 apply(simp add:sign_simps)
nipkow@23482
  1560
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
nipkow@23482
  1561
 prefer 2 apply(simp add:sign_simps)
nipkow@23482
  1562
apply(simp add:field_simps)
avigad@16775
  1563
done
nipkow@23482
  1564
*)
avigad@16775
  1565
wenzelm@23389
  1566
avigad@16775
  1567
subsection{*Division and Signs*}
avigad@16775
  1568
avigad@16775
  1569
lemma zero_less_divide_iff:
avigad@16775
  1570
     "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
  1571
by (simp add: divide_inverse zero_less_mult_iff)
avigad@16775
  1572
avigad@16775
  1573
lemma divide_less_0_iff:
avigad@16775
  1574
     "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
avigad@16775
  1575
      (0 < a & b < 0 | a < 0 & 0 < b)"
avigad@16775
  1576
by (simp add: divide_inverse mult_less_0_iff)
avigad@16775
  1577
avigad@16775
  1578
lemma zero_le_divide_iff:
avigad@16775
  1579
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
avigad@16775
  1580
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
avigad@16775
  1581
by (simp add: divide_inverse zero_le_mult_iff)
avigad@16775
  1582
avigad@16775
  1583
lemma divide_le_0_iff:
avigad@16775
  1584
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
avigad@16775
  1585
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
avigad@16775
  1586
by (simp add: divide_inverse mult_le_0_iff)
avigad@16775
  1587
paulson@24286
  1588
lemma divide_eq_0_iff [simp,noatp]:
avigad@16775
  1589
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
nipkow@23482
  1590
by (simp add: divide_inverse)
avigad@16775
  1591
nipkow@23482
  1592
lemma divide_pos_pos:
nipkow@23482
  1593
  "0 < (x::'a::ordered_field) ==> 0 < y ==> 0 < x / y"
nipkow@23482
  1594
by(simp add:field_simps)
nipkow@23482
  1595
avigad@16775
  1596
nipkow@23482
  1597
lemma divide_nonneg_pos:
nipkow@23482
  1598
  "0 <= (x::'a::ordered_field) ==> 0 < y ==> 0 <= x / y"
nipkow@23482
  1599
by(simp add:field_simps)
avigad@16775
  1600
nipkow@23482
  1601
lemma divide_neg_pos:
nipkow@23482
  1602
  "(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0"
nipkow@23482
  1603
by(simp add:field_simps)
avigad@16775
  1604
nipkow@23482
  1605
lemma divide_nonpos_pos:
nipkow@23482
  1606
  "(x::'a::ordered_field) <= 0 ==> 0 < y ==> x / y <= 0"
nipkow@23482
  1607
by(simp add:field_simps)
avigad@16775
  1608
nipkow@23482
  1609
lemma divide_pos_neg:
nipkow@23482
  1610
  "0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0"
nipkow@23482
  1611
by(simp add:field_simps)
avigad@16775
  1612
nipkow@23482
  1613
lemma divide_nonneg_neg:
nipkow@23482
  1614
  "0 <= (x::'a::ordered_field) ==> y < 0 ==> x / y <= 0" 
nipkow@23482
  1615
by(simp add:field_simps)
avigad@16775
  1616
nipkow@23482
  1617
lemma divide_neg_neg:
nipkow@23482
  1618
  "(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y"
nipkow@23482
  1619
by(simp add:field_simps)
avigad@16775
  1620
nipkow@23482
  1621
lemma divide_nonpos_neg:
nipkow@23482
  1622
  "(x::'a::ordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"
nipkow@23482
  1623
by(simp add:field_simps)
paulson@15234
  1624
wenzelm@23389
  1625
paulson@14288
  1626
subsection{*Cancellation Laws for Division*}
paulson@14288
  1627
paulson@24286
  1628
lemma divide_cancel_right [simp,noatp]:
paulson@14288
  1629
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1630
apply (cases "c=0", simp)
nipkow@23496
  1631
apply (simp add: divide_inverse)
paulson@14288
  1632
done
paulson@14288
  1633
paulson@24286
  1634
lemma divide_cancel_left [simp,noatp]:
paulson@14288
  1635
     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
nipkow@23482
  1636
apply (cases "c=0", simp)
nipkow@23496
  1637
apply (simp add: divide_inverse)
paulson@14288
  1638
done
paulson@14288
  1639
wenzelm@23389
  1640
paulson@14353
  1641
subsection {* Division and the Number One *}
paulson@14353
  1642
paulson@14353
  1643
text{*Simplify expressions equated with 1*}
paulson@24286
  1644
lemma divide_eq_1_iff [simp,noatp]:
paulson@14353
  1645
     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1646
apply (cases "b=0", simp)
nipkow@23482
  1647
apply (simp add: right_inverse_eq)
paulson@14353
  1648
done
paulson@14353
  1649
paulson@24286
  1650
lemma one_eq_divide_iff [simp,noatp]:
paulson@14353
  1651
     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1652
by (simp add: eq_commute [of 1])
paulson@14353
  1653
paulson@24286
  1654
lemma zero_eq_1_divide_iff [simp,noatp]:
paulson@14353
  1655
     "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
nipkow@23482
  1656
apply (cases "a=0", simp)
nipkow@23482
  1657
apply (auto simp add: nonzero_eq_divide_eq)
paulson@14353
  1658
done
paulson@14353
  1659
paulson@24286
  1660
lemma one_divide_eq_0_iff [simp,noatp]:
paulson@14353
  1661
     "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
nipkow@23482
  1662
apply (cases "a=0", simp)
nipkow@23482
  1663
apply (insert zero_neq_one [THEN not_sym])
nipkow@23482
  1664
apply (auto simp add: nonzero_divide_eq_eq)
paulson@14353
  1665
done
paulson@14353
  1666
paulson@14353
  1667
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
paulson@18623
  1668
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
paulson@18623
  1669
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
paulson@18623
  1670
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
paulson@18623
  1671
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]
paulson@17085
  1672
paulson@17085
  1673
declare zero_less_divide_1_iff [simp]
paulson@24286
  1674
declare divide_less_0_1_iff [simp,noatp]
paulson@17085
  1675
declare zero_le_divide_1_iff [simp]
paulson@24286
  1676
declare divide_le_0_1_iff [simp,noatp]
paulson@14353
  1677
wenzelm@23389
  1678
paulson@14293
  1679
subsection {* Ordering Rules for Division *}
paulson@14293
  1680
paulson@14293
  1681
lemma divide_strict_right_mono:
paulson@14293
  1682
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1683
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
nipkow@23482
  1684
              positive_imp_inverse_positive)
paulson@14293
  1685
paulson@14293
  1686
lemma divide_right_mono:
paulson@14293
  1687
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
nipkow@23482
  1688
by (force simp add: divide_strict_right_mono order_le_less)
paulson@14293
  1689
avigad@16775
  1690
lemma divide_right_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1691
    ==> c <= 0 ==> b / c <= a / c"
nipkow@23482
  1692
apply (drule divide_right_mono [of _ _ "- c"])
nipkow@23482
  1693
apply auto
avigad@16775
  1694
done
avigad@16775
  1695
avigad@16775
  1696
lemma divide_strict_right_mono_neg:
avigad@16775
  1697
     "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
nipkow@23482
  1698
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
nipkow@23482
  1699
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])
avigad@16775
  1700
done
paulson@14293
  1701
paulson@14293
  1702
text{*The last premise ensures that @{term a} and @{term b} 
paulson@14293
  1703
      have the same sign*}
paulson@14293
  1704
lemma divide_strict_left_mono:
nipkow@23482
  1705
  "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
nipkow@23482
  1706
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
paulson@14293
  1707
paulson@14293
  1708
lemma divide_left_mono:
nipkow@23482
  1709
  "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
nipkow@23482
  1710
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
paulson@14293
  1711
avigad@16775
  1712
lemma divide_left_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1713
    ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
avigad@16775
  1714
  apply (drule divide_left_mono [of _ _ "- c"])
avigad@16775
  1715
  apply (auto simp add: mult_commute)
avigad@16775
  1716
done
avigad@16775
  1717
paulson@14293
  1718
lemma divide_strict_left_mono_neg:
nipkow@23482
  1719
  "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
nipkow@23482
  1720
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
nipkow@23482
  1721
paulson@14293
  1722
avigad@16775
  1723
text{*Simplify quotients that are compared with the value 1.*}
avigad@16775
  1724
paulson@24286
  1725
lemma le_divide_eq_1 [noatp]:
avigad@16775
  1726
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1727
  shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
avigad@16775
  1728
by (auto simp add: le_divide_eq)
avigad@16775
  1729
paulson@24286
  1730
lemma divide_le_eq_1 [noatp]:
avigad@16775
  1731
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1732
  shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
avigad@16775
  1733
by (auto simp add: divide_le_eq)
avigad@16775
  1734
paulson@24286
  1735
lemma less_divide_eq_1 [noatp]:
avigad@16775
  1736
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1737
  shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
avigad@16775
  1738
by (auto simp add: less_divide_eq)
avigad@16775
  1739
paulson@24286
  1740
lemma divide_less_eq_1 [noatp]:
avigad@16775
  1741
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1742
  shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
avigad@16775
  1743
by (auto simp add: divide_less_eq)
avigad@16775
  1744
wenzelm@23389
  1745
avigad@16775
  1746
subsection{*Conditional Simplification Rules: No Case Splits*}
avigad@16775
  1747
paulson@24286
  1748
lemma le_divide_eq_1_pos [simp,noatp]:
avigad@16775
  1749
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1750
  shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
avigad@16775
  1751
by (auto simp add: le_divide_eq)
avigad@16775
  1752
paulson@24286
  1753
lemma le_divide_eq_1_neg [simp,noatp]:
avigad@16775
  1754
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1755
  shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
avigad@16775
  1756
by (auto simp add: le_divide_eq)
avigad@16775
  1757
paulson@24286
  1758
lemma divide_le_eq_1_pos [simp,noatp]:
avigad@16775
  1759
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1760
  shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
avigad@16775
  1761
by (auto simp add: divide_le_eq)
avigad@16775
  1762
paulson@24286
  1763
lemma divide_le_eq_1_neg [simp,noatp]:
avigad@16775
  1764
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1765
  shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
avigad@16775
  1766
by (auto simp add: divide_le_eq)
avigad@16775
  1767
paulson@24286
  1768
lemma less_divide_eq_1_pos [simp,noatp]:
avigad@16775
  1769
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1770
  shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
avigad@16775
  1771
by (auto simp add: less_divide_eq)
avigad@16775
  1772
paulson@24286
  1773
lemma less_divide_eq_1_neg [simp,noatp]:
avigad@16775
  1774
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1775
  shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
avigad@16775
  1776
by (auto simp add: less_divide_eq)
avigad@16775
  1777
paulson@24286
  1778
lemma divide_less_eq_1_pos [simp,noatp]:
avigad@16775
  1779
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1780
  shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
paulson@18649
  1781
by (auto simp add: divide_less_eq)
paulson@18649
  1782
paulson@24286
  1783
lemma divide_less_eq_1_neg [simp,noatp]:
paulson@18649
  1784
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1785
  shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
avigad@16775
  1786
by (auto simp add: divide_less_eq)
avigad@16775
  1787
paulson@24286
  1788
lemma eq_divide_eq_1 [simp,noatp]:
avigad@16775
  1789
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1790
  shows "(1 = b/a) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1791
by (auto simp add: eq_divide_eq)
avigad@16775
  1792
paulson@24286
  1793
lemma divide_eq_eq_1 [simp,noatp]:
avigad@16775
  1794
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1795
  shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1796
by (auto simp add: divide_eq_eq)
avigad@16775
  1797
wenzelm@23389
  1798
avigad@16775
  1799
subsection {* Reasoning about inequalities with division *}
avigad@16775
  1800
avigad@16775
  1801
lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  1802
    ==> x * y <= x"
avigad@16775
  1803
  by (auto simp add: mult_compare_simps);
avigad@16775
  1804
avigad@16775
  1805
lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  1806
    ==> y * x <= x"
avigad@16775
  1807
  by (auto simp add: mult_compare_simps);
avigad@16775
  1808
avigad@16775
  1809
lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==>
avigad@16775
  1810
    x / y <= z";
avigad@16775
  1811
  by (subst pos_divide_le_eq, assumption+);
avigad@16775
  1812
avigad@16775
  1813
lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==>
nipkow@23482
  1814
    z <= x / y"
nipkow@23482
  1815
by(simp add:field_simps)
avigad@16775
  1816
avigad@16775
  1817
lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==>
avigad@16775
  1818
    x / y < z"
nipkow@23482
  1819
by(simp add:field_simps)
avigad@16775
  1820
avigad@16775
  1821
lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==>
avigad@16775
  1822
    z < x / y"
nipkow@23482
  1823
by(simp add:field_simps)
avigad@16775
  1824
avigad@16775
  1825
lemma frac_le: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  1826
    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
avigad@16775
  1827
  apply (rule mult_imp_div_pos_le)
avigad@16775
  1828
  apply simp;
avigad@16775
  1829
  apply (subst times_divide_eq_left);
avigad@16775
  1830
  apply (rule mult_imp_le_div_pos, assumption)
avigad@16775
  1831
  apply (rule mult_mono)
avigad@16775
  1832
  apply simp_all
paulson@14293
  1833
done
paulson@14293
  1834
avigad@16775
  1835
lemma frac_less: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  1836
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
avigad@16775
  1837
  apply (rule mult_imp_div_pos_less)
avigad@16775
  1838
  apply simp;
avigad@16775
  1839
  apply (subst times_divide_eq_left);
avigad@16775
  1840
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  1841
  apply (erule mult_less_le_imp_less)
avigad@16775
  1842
  apply simp_all
avigad@16775
  1843
done
avigad@16775
  1844
avigad@16775
  1845
lemma frac_less2: "(0::'a::ordered_field) < x ==> 
avigad@16775
  1846
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
avigad@16775
  1847
  apply (rule mult_imp_div_pos_less)
avigad@16775
  1848
  apply simp_all
avigad@16775
  1849
  apply (subst times_divide_eq_left);
avigad@16775
  1850
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  1851
  apply (erule mult_le_less_imp_less)
avigad@16775
  1852
  apply simp_all
avigad@16775
  1853
done
avigad@16775
  1854
avigad@16775
  1855
text{*It's not obvious whether these should be simprules or not. 
avigad@16775
  1856
  Their effect is to gather terms into one big fraction, like
avigad@16775
  1857
  a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
avigad@16775
  1858
  seem to need them.*}
avigad@16775
  1859
avigad@16775
  1860
declare times_divide_eq [simp]
paulson@14293
  1861
wenzelm@23389
  1862
paulson@14293
  1863
subsection {* Ordered Fields are Dense *}
paulson@14293
  1864
obua@14738
  1865
lemma less_add_one: "a < (a+1::'a::ordered_semidom)"
paulson@14293
  1866
proof -
obua@14738
  1867
  have "a+0 < (a+1::'a::ordered_semidom)"
nipkow@23482
  1868
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  1869
  thus ?thesis by simp
paulson@14293
  1870
qed
paulson@14293
  1871
obua@14738
  1872
lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)"
nipkow@23482
  1873
by (blast intro: order_less_trans zero_less_one less_add_one)
paulson@14365
  1874
paulson@14293
  1875
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
nipkow@23482
  1876
by (simp add: field_simps zero_less_two)
paulson@14293
  1877
paulson@14293
  1878
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
nipkow@23482
  1879
by (simp add: field_simps zero_less_two)
paulson@14293
  1880
paulson@14293
  1881
lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b"
paulson@14293
  1882
by (blast intro!: less_half_sum gt_half_sum)
paulson@14293
  1883
paulson@15234
  1884
paulson@14293
  1885
subsection {* Absolute Value *}
paulson@14293
  1886
obua@14738
  1887
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
nipkow@23477
  1888
by (simp add: abs_if zero_less_one [THEN order_less_not_sym])
paulson@14294
  1889
obua@14738
  1890
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 
obua@14738
  1891
proof -
obua@14738
  1892
  let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
obua@14738
  1893
  let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
obua@14738
  1894
  have a: "(abs a) * (abs b) = ?x"
nipkow@23477
  1895
    by (simp only: abs_prts[of a] abs_prts[of b] ring_simps)
obua@14738
  1896
  {
obua@14738
  1897
    fix u v :: 'a
paulson@15481
  1898
    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> 
paulson@15481
  1899
              u * v = pprt a * pprt b + pprt a * nprt b + 
paulson@15481
  1900
                      nprt a * pprt b + nprt a * nprt b"
obua@14738
  1901
      apply (subst prts[of u], subst prts[of v])
nipkow@23477
  1902
      apply (simp add: ring_simps) 
obua@14738
  1903
      done
obua@14738
  1904
  }
obua@14738
  1905
  note b = this[OF refl[of a] refl[of b]]
obua@14738
  1906
  note addm = add_mono[of "0::'a" _ "0::'a", simplified]
obua@14738
  1907
  note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
obua@14738
  1908
  have xy: "- ?x <= ?y"
obua@14754
  1909
    apply (simp)
obua@14754
  1910
    apply (rule_tac y="0::'a" in order_trans)
nipkow@16568
  1911
    apply (rule addm2)
avigad@16775
  1912
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
nipkow@16568
  1913
    apply (rule addm)
avigad@16775
  1914
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
obua@14754
  1915
    done
obua@14738
  1916
  have yx: "?y <= ?x"
nipkow@16568
  1917
    apply (simp add:diff_def)
obua@14754
  1918
    apply (rule_tac y=0 in order_trans)
avigad@16775
  1919
    apply (rule addm2, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
avigad@16775
  1920
    apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
obua@14738
  1921
    done
obua@14738
  1922
  have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
obua@14738
  1923
  have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
obua@14738
  1924
  show ?thesis
obua@14738
  1925
    apply (rule abs_leI)
obua@14738
  1926
    apply (simp add: i1)
obua@14738
  1927
    apply (simp add: i2[simplified minus_le_iff])
obua@14738
  1928
    done
obua@14738
  1929
qed
paulson@14294
  1930
obua@14738
  1931
lemma abs_eq_mult: 
obua@14738
  1932
  assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
obua@14738
  1933
  shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"
obua@14738
  1934
proof -
obua@14738
  1935
  have s: "(0 <= a*b) | (a*b <= 0)"
obua@14738
  1936
    apply (auto)    
obua@14738
  1937
    apply (rule_tac split_mult_pos_le)
obua@14738
  1938
    apply (rule_tac contrapos_np[of "a*b <= 0"])
obua@14738
  1939
    apply (simp)
obua@14738
  1940
    apply (rule_tac split_mult_neg_le)
obua@14738
  1941
    apply (insert prems)
obua@14738
  1942
    apply (blast)
obua@14738
  1943
    done
obua@14738
  1944
  have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
obua@14738
  1945
    by (simp add: prts[symmetric])
obua@14738
  1946
  show ?thesis
obua@14738
  1947
  proof cases
obua@14738
  1948
    assume "0 <= a * b"
obua@14738
  1949
    then show ?thesis
obua@14738
  1950
      apply (simp_all add: mulprts abs_prts)
obua@14738
  1951
      apply (insert prems)
obua@14754
  1952
      apply (auto simp add: 
nipkow@23477
  1953
	ring_simps 
obua@14754
  1954
	iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]
nipkow@15197
  1955
	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id])
avigad@16775
  1956
	apply(drule (1) mult_nonneg_nonpos[of a b], simp)
avigad@16775
  1957
	apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
obua@14738
  1958
      done
obua@14738
  1959
  next
obua@14738
  1960
    assume "~(0 <= a*b)"
obua@14738
  1961
    with s have "a*b <= 0" by simp
obua@14738
  1962
    then show ?thesis
obua@14738
  1963
      apply (simp_all add: mulprts abs_prts)
obua@14738
  1964
      apply (insert prems)
nipkow@23477
  1965
      apply (auto simp add: ring_simps)
avigad@16775
  1966
      apply(drule (1) mult_nonneg_nonneg[of a b],simp)
avigad@16775
  1967
      apply(drule (1) mult_nonpos_nonpos[of a b],simp)
obua@14738
  1968
      done
obua@14738
  1969
  qed
obua@14738
  1970
qed
paulson@14294
  1971
obua@14738
  1972
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" 
obua@14738
  1973
by (simp add: abs_eq_mult linorder_linear)
paulson@14293
  1974
obua@14738
  1975
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
obua@14738
  1976
by (simp add: abs_if) 
paulson@14294
  1977
paulson@14294
  1978
lemma nonzero_abs_inverse:
paulson@14294
  1979
     "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
paulson@14294
  1980
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
paulson@14294
  1981
                      negative_imp_inverse_negative)
paulson@14294
  1982
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
paulson@14294
  1983
done
paulson@14294
  1984
paulson@14294
  1985
lemma abs_inverse [simp]:
paulson@14294
  1986
     "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
paulson@14294
  1987
      inverse (abs a)"
haftmann@21328
  1988
apply (cases "a=0", simp) 
paulson@14294
  1989
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  1990
done
paulson@14294
  1991
paulson@14294
  1992
lemma nonzero_abs_divide:
paulson@14294
  1993
     "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
paulson@14294
  1994
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
paulson@14294
  1995
paulson@15234
  1996
lemma abs_divide [simp]:
paulson@14294
  1997
     "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
haftmann@21328
  1998
apply (cases "b=0", simp) 
paulson@14294
  1999
apply (simp add: nonzero_abs_divide) 
paulson@14294
  2000
done
paulson@14294
  2001
paulson@14294
  2002
lemma abs_mult_less:
obua@14738
  2003
     "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
paulson@14294
  2004
proof -
paulson@14294
  2005
  assume ac: "abs a < c"
paulson@14294
  2006
  hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
paulson@14294
  2007
  assume "abs b < d"
paulson@14294
  2008
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  2009
qed
paulson@14293
  2010
obua@14738
  2011
lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))"
obua@14738
  2012
by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)
obua@14738
  2013
obua@14738
  2014
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
obua@14738
  2015
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)
obua@14738
  2016
obua@14738
  2017
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 
obua@14738
  2018
apply (simp add: order_less_le abs_le_iff)  
obua@14738
  2019
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff)
obua@14738
  2020
apply (simp add: le_minus_self_iff linorder_neq_iff) 
obua@14738
  2021
done
obua@14738
  2022
avigad@16775
  2023
lemma abs_mult_pos: "(0::'a::ordered_idom) <= x ==> 
avigad@16775
  2024
    (abs y) * x = abs (y * x)";
avigad@16775
  2025
  apply (subst abs_mult);
avigad@16775
  2026
  apply simp;
avigad@16775
  2027
done;
avigad@16775
  2028
avigad@16775
  2029
lemma abs_div_pos: "(0::'a::{division_by_zero,ordered_field}) < y ==> 
avigad@16775
  2030
    abs x / y = abs (x / y)";
avigad@16775
  2031
  apply (subst abs_divide);
avigad@16775
  2032
  apply (simp add: order_less_imp_le);
avigad@16775
  2033
done;
avigad@16775
  2034
wenzelm@23389
  2035
obua@19404
  2036
subsection {* Bounds of products via negative and positive Part *}
obua@15178
  2037
obua@15580
  2038
lemma mult_le_prts:
obua@15580
  2039
  assumes
obua@15580
  2040
  "a1 <= (a::'a::lordered_ring)"
obua@15580
  2041
  "a <= a2"
obua@15580
  2042
  "b1 <= b"
obua@15580
  2043
  "b <= b2"
obua@15580
  2044
  shows
obua@15580
  2045
  "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
obua@15580
  2046
proof - 
obua@15580
  2047
  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" 
obua@15580
  2048
    apply (subst prts[symmetric])+
obua@15580
  2049
    apply simp
obua@15580
  2050
    done
obua@15580
  2051
  then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
nipkow@23477
  2052
    by (simp add: ring_simps)
obua@15580
  2053
  moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
obua@15580
  2054
    by (simp_all add: prems mult_mono)
obua@15580
  2055
  moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
obua@15580
  2056
  proof -
obua@15580
  2057
    have "pprt a * nprt b <= pprt a * nprt b2"
obua@15580
  2058
      by (simp add: mult_left_mono prems)
obua@15580
  2059
    moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
obua@15580
  2060
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2061
    ultimately show ?thesis
obua@15580
  2062
      by simp
obua@15580
  2063
  qed
obua@15580
  2064
  moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
obua@15580
  2065
  proof - 
obua@15580
  2066
    have "nprt a * pprt b <= nprt a2 * pprt b"
obua@15580
  2067
      by (simp add: mult_right_mono prems)
obua@15580
  2068
    moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
obua@15580
  2069
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2070
    ultimately show ?thesis
obua@15580
  2071
      by simp
obua@15580
  2072
  qed
obua@15580
  2073
  moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
obua@15580
  2074
  proof -
obua@15580
  2075
    have "nprt a * nprt b <= nprt a * nprt b1"
obua@15580
  2076
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2077
    moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
obua@15580
  2078
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2079
    ultimately show ?thesis
obua@15580
  2080
      by simp
obua@15580
  2081
  qed
obua@15580
  2082
  ultimately show ?thesis
obua@15580
  2083
    by - (rule add_mono | simp)+
obua@15580
  2084
qed
obua@19404
  2085
obua@19404
  2086
lemma mult_ge_prts:
obua@15178
  2087
  assumes
obua@19404
  2088
  "a1 <= (a::'a::lordered_ring)"
obua@19404
  2089
  "a <= a2"
obua@19404
  2090
  "b1 <= b"
obua@19404
  2091
  "b <= b2"
obua@15178
  2092
  shows
obua@19404
  2093
  "a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
obua@19404
  2094
proof - 
obua@19404
  2095
  from prems have a1:"- a2 <= -a" by auto
obua@19404
  2096
  from prems have a2: "-a <= -a1" by auto
obua@19404
  2097
  from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 prems(3) prems(4), simplified nprt_neg pprt_neg] 
obua@19404
  2098
  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp  
obua@19404
  2099
  then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
obua@19404
  2100
    by (simp only: minus_le_iff)
obua@19404
  2101
  then show ?thesis by simp
obua@15178
  2102
qed
obua@15178
  2103
wenzelm@23389
  2104
haftmann@22842
  2105
subsection {* Theorems for proof tools *}
haftmann@22842
  2106
haftmann@22842
  2107
lemma add_mono_thms_ordered_semiring:
haftmann@22842
  2108
  fixes i j k :: "'a\<Colon>pordered_ab_semigroup_add"
haftmann@22842
  2109
  shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@22842
  2110
    and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@22842
  2111
    and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
haftmann@22842
  2112
    and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
haftmann@22842
  2113
by (rule add_mono, clarify+)+
haftmann@22842
  2114
haftmann@22842
  2115
lemma add_mono_thms_ordered_field:
haftmann@22842
  2116
  fixes i j k :: "'a\<Colon>pordered_cancel_ab_semigroup_add"
haftmann@22842
  2117
  shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2118
    and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2119
    and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2120
    and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2121
    and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2122
by (auto intro: add_strict_right_mono add_strict_left_mono
haftmann@22842
  2123
  add_less_le_mono add_le_less_mono add_strict_mono)
haftmann@22842
  2124
paulson@14265
  2125
end